SAT Area and Volume Questions - Real Collegeboard

Choice D is correct. The perimeter of a figure is equal to the sum of the measurements of the sides of the figure. It’s given that the rectangle has a length of \(4\) inches and a width of \(9\) inches. Since a rectangle has \(4\) sides, of which opposite sides are parallel and equal, it follows that the rectangle has two sides with a length of \(4\) inches and two sides with a width of \(9\) inches. Therefore, the perimeter of this rectangle is \(4 + 4 + 9 + 9\), or \(26\) inches. Choice A is incorrect. This is the sum, in inches, of the length and the width of the rectangle. Choice B is incorrect. This is the sum, in inches, of the two lengths and the width of the rectangle. Choice C is incorrect. This is the sum, in inches, of the length and the two widths of the rectangle.

Choice C is correct. The two diagonals of square ABCD divide the square into 4 congruent right triangles, where each triangle has a vertex at the origin of the graph shown. The formula for the area of a triangle is, where b is the base length of the triangle and h is the height of the triangle. Each of the 4 congruent right triangles has a height of 5 units and a base length of 5 units. Therefore, the area of each triangle is, or 12.5 square units. Since the 4 right triangles are congruent, the area of each is of the area of square ABCD. It follows that the area of the square ABCD is equal to, or 50 square units.Choices A and D are incorrect and may result from using 5 or 25, respectively, as the area of one of the 4 congruent right triangles formed by diagonals of square ABCD. However, the area of these triangles is 12.5. Choice B is incorrect and may result from using 5 as the length of one side of square ABCD. However, the length of a side of square ABCD is .

Choice B is correct. The surface area of a cube with side length s is equal to . Since the surface area is given as 54 square meters, the equation can be used to solve for s. Dividing both sides of the equation by 6 yields . Taking the square root of both sides of this equation yields and . Since the side length of a cube must be a positive value, can be discarded as a possible solution, leaving . The volume of a cube with side length s is equal to . Therefore, the volume of this cube, in cubic meters, is, or 27.Choices A, C, and D are incorrect and may result from calculation errors.

Choice A is correct. It’s given that the volume of a right rectangular prism is . The prism shown has a length of 6 cm, a width of 2 cm, and a height of 3 cm. Thus,, or 36 cubic centimeters.Choice B is incorrect. This is the volume of a rectangular prism with edge lengths of 6, 2, and 2. Choice C is incorrect and may result from only finding the product of the length and width of the base of the prism. Choice D is incorrect and may result from finding the sum, not the product, of the edge lengths of the prism.

The correct answer is \(135/8\). It's given that triangles \( A B C\) and \( D E F\) are similar and each side length of triangle \( A B C\) is \(4\) times the corresponding side length of triangle \( D E F\). For two similar triangles, if each side length of the first triangle is \(k\) times the corresponding side length of the second triangle, then the area of the first triangle is \(k ^2\) times the area of the second triangle. Therefore, the area of triangle \( A B C\) is \(4 ^2\), or \(16\), times the area of triangle \( D E F\). It's given that the area of triangle \( A B C\) is \(270\) square inches. Let \(a\) represent the area, in square inches, of triangle \( D E F\). It follows that \(270\) is \(16\) times \(a\), or \(270 = 16 a\). Dividing both sides of this equation by \(16\) yields \(270/16 = a\), which is equivalent to \(135/8 = a\). Thus, the area, in square inches, of triangle \( D E F\) is \(135/8\). Note that 135/8, 16.87, and 16.88 are examples of ways to enter a correct answer.

Choice D is correct. Using the volume formula and the given information that the volume of the glass is 473 cubic centimeters, the value of k can be found as follows:Therefore, the value of k is approximately 10.11 centimeters.Choices A, B, and C are incorrect. Substituting the values of k from these choices in the formula results in volumes of approximately 7 cubic centimeters, 207 cubic centimeters, and 217 cubic centimeters, respectively, all of which contradict the given information that the volume of the glass is 473 cubic centimeters.

Choice B is correct. The area, \(A\), of a triangle can be found using the formula \(A = 1 half b h\), where \(b\) is the length of the base of the triangle and \(h\) is the height of the triangle. It's given that the triangle is a right triangle. Therefore, its base and height can be represented by the two legs. It’s also given that the triangle has sides of length \(2 √2 \), \(6 √2 \), and \(√80 \) units. Since \(√80 \) units is the greatest of these lengths, it's the length of the hypotenuse. Therefore, the two legs have lengths \(2 √2 \) and \(6 √2 \) units. Substituting these values for \(b\) and \(h\) in the formula \(A = 1 half b h\) gives \(A = 1 half(2 √2 )(6 √2 )\), which is equivalent to \(A = 6 √4 \) square units, or \(A = 12\) square units. Choice A is incorrect. This expression represents the perimeter, rather than the area, of the triangle. Choice C is incorrect and may result from conceptual or calculation errors. Choice D is incorrect and may result from conceptual or calculation errors.

The correct answer is \(2,048\). The area \(A\), in square inches, of a rectangle is equal to the product of its length \(ell\), in inches, and its width \(w\), in inches, or \(A = ell w\). It's given that the rectangle has a length of \(64\) inches and a width of \(32\) inches. Substituting \(64\) for \(ell\) and \(32\) for \(w\) in the equation \(A = ell w\) yields \(A =(64)(32)\), or \(A = 2,048\). Therefore, the area, in square inches, of the rectangle is \(2,048\).

Choice C is correct. The total volume of the cylindrical can is found by multiplying the area of the base of the can,, by the height of the can, 10 cm, which yields . If the syrup needed to fill the can has a volume of, then the remaining volume for theπeces offruit is .Choice A is incorrect because if the fruit had a volume of, there would be of syrup needed to fill the can to the top. Choice B is incorrect because if the fruit had a volume of, there would be of syrup needed to fill the can to the top. Choice D is incorrect because it is the total volume of the can, not just of theπeces of fruit.

Choice A is correct. The volume, \( V\), of a right circular cylinder is given by the formula \( V =π r ^2 h\), where \(π r ^2\) is the area of the base of the cylinder and \(h\) is the height. It’s given that a right circular cylinder has a volume of \(432\) cubic centimeters and the area of the base is \(24\) square centimeters. Substituting \(432\) for \( V\) and \(24\) for \(π r ^2\) in the formula \( V =π r ^2 h\) yields \(432 = 24 h\). Dividing both sides of this equation by \(24\) yields \(18 = h\). Therefore, the height of the cylinder, in centimeters, is \(18\). Choice B is incorrect. This is the area of the base, in square centimeters, not the height, in centimeters, of the cylinder. Choice C is incorrect. This is the height, in centimeters, of a cylinder if its volume is \(432\) cubic centimeters and the area of its base is \(2\), not \(24\), cubic centimeters. Choice D is incorrect. This is the height, in centimeters, of a cylinder if its volume is \(432\) cubic centimeters and the area of its base is \(1 20 fourth\), not \(24\), cubic centimeters.

Choice D is correct. The surface area is found by summing the area of each face. A right rectangular prism consists of three pairs of congruent rectangles, so the surface area is found by multiplying the areas of three adjacent rectangles by 2 and adding these products. For this prism, the surface area is equal to, or, which is equal to 148.Choice A is incorrect. This is the area of one of the faces of the prism. Choice B is incorrect and may result from adding the areas of three adjacent rectangles without multiplying by 2. Choice C is incorrect. This is the volume, in cubic inches, of the prism.

The correct answer is \(180\). The perimeter of a polygon is equal to the sum of the lengths of the sides of the polygon. It’s given that each side of the square has a length of \(45\). Since a square is a polygon with \(4\) sides, the perimeter of this square is \(45 + 45 + 45 + 45\), or \(180\).

Choice B is correct. The length of a segment in the xy-plane can be found using the distance formula, \(√(x_2 -x_1)^2 +(y_2 -y_1)^2 \), where \((x_1, y_1)\) and \((x_2, y_2)\) are the endpoints of the segment. The segment shown has endpoints at \((-6, 4)\) and \((3, 10)\). Substituting \((-6, 4)\) and \((3, 10)\) for \((x_1, y_1)\) and \((x_2, y_2)\), respectively, in the distance formula yields \(√(3 -(-6))^2 +(10 -4)^2 \), or \(√9 ^2 + 6 ^2 \), which is equivalent to \(√81 + 36 \), or \(√117 \). Let \(x\) represent the length, in units, of the other leg of this triangle. The area, \(A\), of a right triangle can be calculated using the formula \(A = 1 half b h\), where \(b\) and \(h\) are the lengths of the legs of the triangle. It's given that the area of the triangle is \(36 √13 \) square units. Substituting \(36 √13 \) for \(A\), \(√117 \) for \(b\), and \(x\) for \(h\) in the formula \(A = 1 half b h\) yields \(36 √13 = 1 half(√117 )(x)\). Multiplying both sides of this equation by \(2\) yields \(72 √13 = x √117 \). Dividing both sides of this equation by \(√117 \) yields \(72 √13 /√117 = x\). Multiplying the numerator and denominator on the left-hand side of this equation by \(√117 \) yields \(72 √1,521 /117 = x\), or \(72(39)/117 = x\), which is equivalent to \(2,808/117 = x\), or \(24 = x\). Therefore, the length, in units, of the other leg of this triangle is \(24\). Choice A is incorrect and may result from conceptual or calculation errors. Choice C is incorrect. \(3 √13 \) is equivalent to \(√117 \), which is the length, in units, of the line segment shown in the xy-plane, not the length, in units, of the other leg of the triangle. Choice D is incorrect and may result from conceptual or calculation errors.

Choice B is correct. The area of a rectangle with length \(ell\) and width \(w\) can be found using the formula \(A = ell w\). It’s given that the rectangle has a length of \(4 cm\) and a width of \(2 cm\). Therefore, the area of this rectangle is \((4 cm)(2 cm)\), or \(8 cm ^2\). Choice A is incorrect. This is the sum, \(in cm\), of the length and width of the rectangle, not the area, \(in cm ^2\). Choice C is incorrect. This is the perimeter, \(in cm\), of the rectangle, not the area, \(in cm ^2\). Choice D is incorrect. This is the sum of the length and width of the rectangle ^2, not the area.

Choice C is correct. The volume, \( V\), of a right circular cylinder is given by the formula \( V =π r ^2 h\), where \(r\) is the radius of the base of the cylinder and \(h\) is the height of the cylinder. It's given that a right circular cylinder has a height of \(6\) centimeters. Therefore, \(h = 6\). It's also given that the cylinder has a base diameter of \(22\) centimeters. The radius of a circle is half the diameter of the circle. Since the base of a right circular cylinder is a circle, it follows that the radius of the base of the right circular cylinder is \(22/2\), or \(11\), centimeters. Therefore, \(r = 11\). Substituting \(11\) for \(r\) and \(6\) for \(h\) in the formula \( V =π r ^2 h\) yields \( V =π(11)^2(6)\), which is equivalent to \( V =π(121)(6)\), or \( V = 726π\). Therefore, the volume, in cubic centimeters, of the cylinder is \(726π\). Choice A is incorrect. This is the volume of a right circular cylinder that has a base diameter of \(2 √22 \), not \(22\), centimeters and a height of \(6\) centimeters. Choice B is incorrect. This is the volume of a right circular cylinder that has a base diameter of \(4 √11 \), not \(22\), centimeters and a height of \(6\) centimeters. Choice D is incorrect. This is the volume of a right circular cylinder that has a base diameter of \(44\), not \(22\), centimeters and a height of \(6\) centimeters.

The correct answer is \(1,800\). The area, \(A\), of a triangle can be found using the formula \(A = 1 half b h\), where \(b\) is the base length of the triangle and \(h\) is the height of the triangle. It’s given that the triangle has a base length of \(40\) centimeters and a height of \(90\) centimeters. Substituting \(40\) for \(b\) and \(90\) for \(h\) in the formula \(A = 1 half b h\) yields \(A = 1 half(40)(90)\), or \(A = 1,800\). Therefore, the area, in square centimeters, of the triangle is \(1,800\).

Choice D is correct. The area of a circle can be found by using the formula \(A =π r ^2\), where \(A\) is the area and \(r\) is the radius of the circle. It’s given that the radius of circle A is \(3 n\). Substituting this value for \(r\) into the formula \(A =π r ^2\) gives \(A =π(3 n)^2\), or \(9π n ^2\). It’s also given that the radius of circle B is \(129 n\). Substituting this value for \(r\) into the formula \(A =π r ^2\) gives \(A =π(129 n)^2\), or \(16,641π n ^2\). Dividing the area of circle B by the area of circle A gives \(16,641π n ^2/9π n ^2\), which simplifies to \(1,849\). Therefore, the area of circle B is \(1,849\) times the area of circle A. Choice A is incorrect. This is how many times greater the radius of circle B is than the radius of circle A. Choice B is incorrect and may result from conceptual or calculation errors. Choice C is incorrect. This is the coefficient on the term that describes the radius of circle B.

The correct answer is \(24.5\). It's given that a triangle is formed by connecting the three points shown, which are \((-3, 4)\), \((5, 3)\), and \((4, -3)\). Let this triangle be triangle A. The area of triangle A can be found by calculating the area of the rectangle that circumscribes it and subtracting the areas of the three triangles that are inside the rectangle but outside triangle A. The rectangle formed by the points \((-3, 4)\), \((5, 4)\), \((5, -3)\), and \((-3, -3)\) circumscribes triangle A. The width, in units, of this rectangle can be found by calculating the distance between the points \((5, 4)\) and \((5, -3)\). This distance is \(4 -(-3)\), or \(7\). The length, in units, of this rectangle can be found by calculating the distance between the points \((5, 4)\) and \((-3, 4)\). This distance is \(5 -(-3)\), or \(8\). It follows that the area, in square units, of the rectangle is \((7)(8)\), or \(56\). One of the triangles that lies inside the rectangle but outside triangle A is formed by the points \((-3, 4)\), \((5, 4)\), and \((5, 3)\). The length, in units, of a base of this triangle can be found by calculating the distance between the points \((5, 4)\) and \((5, 3)\). This distance is \(4 -3\), or \(1\). The corresponding height, in units, of this triangle can be found by calculating the distance between the points \((5, 4)\) and \((-3, 4)\). This distance is \(5 -(-3)\), or \(8\). It follows that the area, in square units, of this triangle is \(1 half(8)(1)\), or \(4\). A second triangle that lies inside the rectangle but outside triangle A is formed by the points \((4, -3)\), \((5, 3)\), and \((5, -3)\). The length, in units, of a base of this triangle can be found by calculating the distance between the points \((5, 3)\) and \((5, -3)\). This distance is \(3 -(-3)\), or \(6\). The corresponding height, in units, of this triangle can be found by calculating the distance between the points \((5, -3)\) and \((4, -3)\). This distance is \(5 -4\), or \(1\). It follows that the area, in square units, of this triangle is \(1 half(1)(6)\), or \(3\). The third triangle that lies inside the rectangle but outside triangle A is formed by the points \((-3, 4)\), \((-3, -3)\), and \((4, -3)\). The length, in units, of a base of this triangle can be found by calculating the distance between the points \((4, -3)\) and \((-3, -3)\). This distance is \(4 -(-3)\), or \(7\). The corresponding height, in units, of this triangle can be found by calculating the distance between the points \((-3, 4)\) and \((-3, -3)\). This distance is \(4 -(-3)\), or \(7\). It follows that the area, in square units, of this triangle is \(1 half(7)(7)\), or \(24.5\). Thus, the area, in square units, of the triangle formed by connecting the three points shown is \(56 -4 -3 -24.5\), or \(24.5\). Note that 24.5 and 49/2 are examples of ways to enter a correct answer.

The correct answer is \(3\). The area of a rectangle can be calculated by multiplying the length of its longest side by the length of its shortest side. It’s given that the area of the rectangle is \(57\) square inches and the length of the longest side of the rectangle is \(19\) inches. Let \(x\) represent the length, in inches, of the shortest side of this rectangle. It follows that \(57 = 19 x\). Dividing both sides of this equation by \(19\) yields \(3 = x\). Therefore, the length, in inches, of the shortest side of the rectangle is \(3\).

Choice C is correct. The area of a rectangle with length \(l\) and width \(w\) can be found using the formula \(A = l w\). It’s given that the rectangle has a length of \(17 cm\) and a width of \(7 cm\). Therefore, the area of this rectangle is \(A = 17(7)\), or \(119 cm ^2\). Choice A is incorrect. This is the sum of the length and width of the rectangle, not the area. Choice B is incorrect. This is the perimeter of the rectangle, not the area. Choice D is incorrect. This is the sum of the length and width of the rectangle ^2, not the area.

Choice A is correct. The area \(A\), in square inches, of a rectangle is the product of its length \(ell\), in inches, and its width \(w\), in inches; thus, \(A = ell w\). It's given that the area of a rectangle is \(630\) square inches and the length of the rectangle is \(70\) inches. Substituting \(630\) for \(A\) and \(70\) for \(ell\) in the equation \(A = ell w\) yields \(630 = 70 w\). Dividing both sides of this equation by \(70\) yields \(9 = w\). Therefore, the width, in inches, of this rectangle is \(9\). Choice B is incorrect. This is the length, not the width, in inches, of the rectangle. Choice C is incorrect. This is half the area, in square inches, not the width, in inches, of the rectangle. Choice D is incorrect. This is the difference between the area, in square inches, and the length, in inches, of the rectangle, not the width, in inches, of the rectangle.

Choice C is correct. The volume of right circular cylinder A is given by the expression, where r is the radius of its circular base and h is its height. The volume of a cylinder with twice the radius and half the height of cylinder A is given by, which is equivalent to . Therefore, the volume is twice the volume of cylinder A, or .Choice A is incorrect and likely results from not multiplying the radius of cylinder A by 2. Choice B is incorrect and likely results from not squaring the 2 in 2r when applying the volume formula. Choice D is incorrect and likely results from a conceptual error.

Choice B is correct. Let \(x\) represent the side length, in \(cm\), of each square base. If the two prisms are glued together along a square base, the resulting prism has a surface area equal to twice the surface area of one of the prisms, minus the area of the two square bases that are being glued together, which yields \(2 K -2 x ^2 cm ^2\) . It’s given that this resulting surface area is equal to \(92/47 K cm ^2\), so \(2 K -2 x ^2 = 92/47 K\). Subtracting \(92/47 K\) from both sides of this equation yields \(2 K -92/47 K -2 x ^2 = 0\). This equation can be rewritten by multiplying \(2 K\) on the left-hand side by \(47/47\), which yields \(94/47 K -92/47 K -2 x ^2 = 0\), or \(2 40 sevenths K -2 x ^2 = 0\). Adding \(2 x ^2\) to both sides of this equation yields \(2 40 sevenths K = 2 x ^2\). Multiplying both sides of this equation by \(47/2\) yields \( K = 47 x ^2\). The surface area \( K\), in \(cm Superscript 2\), of each rectangular prism is equivalent to the sum of the areas of the two square bases and the areas of the four lateral faces. Since the height of each rectangular prism is \(90 cm\) and the side length of each square base is \(x cm\), it follows that the area of each square base is \(x ^2 cm ^2\) and the area of each lateral face is \(90 x cm ^2\). Therefore, the surface area of each rectangular prism can be represented by the expression \(2 x ^2 + 4(90 x)\), or \(2 x ^2 + 360 x\). Substituting this expression for \( K\) in the equation \( K = 47 x ^2\) yields \(2 x ^2 + 360 x = 47 x ^2\). Subtracting \(2 x ^2\) and \(360 x\) from both sides of this equation yields \(0 = 45 x ^2 -360 x\). Factoring \(x\) from the right-hand side of this equation yields \(0 = x(45 x -360)\). Applying the zero product property, it follows that \(x = 0\) and \(45 x -360 = 0\). Adding \(360\) to both sides of the equation \(45 x -360 = 0\) yields \(45 x = 360\). Dividing both sides of this equation by \(45\) yields \(x = 8\). Since a side length of a rectangular prism can’t be \(0\), the length of each square base is \(8\) \(cm\). Choice A is incorrect and may result from conceptual or calculation errors. Choice C is incorrect and may result from conceptual or calculation errors. Choice D is incorrect and may result from conceptual or calculation errors.

Choice D is correct. The area \(A\), in square centimeters, of a rectangle can be found using the formula \(A = ell w\), where \(ell\) is the length, in centimeters, of the rectangle and \(w\) is its width, in centimeters. It's given that the rectangle has a length of \(36\) centimeters and a width of \(34\) centimeters. Substituting \(36\) for \(ell\) and \(34\) for \(w\) in the formula \(A = ell w\) yields \(A = 36(34)\), or \(A = 1,224\). Therefore, the area, in square centimeters, of this rectangle is \(1,224\). Choice A is incorrect and may result from conceptual or calculation errors. Choice B is incorrect. This is the perimeter, in centimeters, not the area, in square centimeters, of the rectangle. Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is correct. The volume, \( V\), of a right circular cone is given by the formula \( V = 1 thirdπ r ^2 h\), where \(π r ^2\) is the area of the circular base of the cone and \(h\) is the height. It’s given that this right circular cone has a volume of \(71,148π\) cubic centimeters and the area of its base is \(5,929π\) square centimeters. Substituting \(71,148π\) for \( V\) and \(5,929π\) for \(π r ^2\) in the formula \( V = 1 thirdπ r ^2 h\) yields \(71,148π =(1 third)(5,929π)(h)\). Dividing each side of this equation by \(5,929π\) yields \(12 = h/3\). Multiplying each side of this equation by \(3\) yields \(36 = h\). Let \(s\) represent the slant height, in centimeters, of this cone. A right triangle is formed by the radius, \(r\), height, \(h\), and slant height, \(s\), of this cone, where \(r\) and \(h\) are the legs of the triangle and \(s\) is the hypotenuse. Using the Pythagorean theorem, the equation \(r ^2 + h ^2 = s ^2\) represents this relationship. Because \(5,929π\) is the area of the base and the area of the base is \(π r ^2\), it follows that \(5,929π =π r ^2\). Dividing both sides of this equation by \(π\) yields \(5,929 = r ^2\). Substituting \(5,929\) for \(r ^2\) and \(36\) for \(h\) in the equation \(r ^2 + h ^2 = s ^2\) yields \(5,929 + 36 ^2 = s ^2\), which is equivalent to \(5,929 + 1,296 = s ^2\), or \(7,225 = s ^2\). Taking the positive square root of both sides of this equation yields \(85 = s\). Therefore, the slant height of the cone is \(85\) centimeters. Choice A is incorrect. This is one-third of the height, in centimeters, not the slant height, in centimeters, of this cone. Choice B is incorrect. This is the height, in centimeters, not the slant height, in centimeters, of this cone. Choice C is incorrect. This is the radius, in centimeters, of the base, not the slant height, in centimeters, of this cone.

Choice A is correct. The perimeter of a triangle is the sum of the lengths of its three sides. The triangle shown has side lengths \(x\), \(y\), and \(z\). It's given that the triangle has a perimeter of \(22\) units. Therefore, \(x + y + z = 22\). If \(x = 9\) units and \(y = 7\) units, the value of \(z\), in units, can be found by substituting \(9\) for \(x\) and \(7\) for \(y\) in the equation \(x + y + z = 22\), which yields \(9 + 7 + z = 22\), or \(16 + z = 22\). Subtracting \(16\) from both sides of this equation yields \(z = 6\). Therefore, if \(x = 9\) units and \(y = 7\) units, the value of \(z\), in units, is \(6\). Choice B is incorrect. This is the value of \(y\), in units, not the value of \(z\), in units. Choice C is incorrect. This is the value of \(x\), in units, not the value of \(z\), in units. Choice D is incorrect. This is the value of \(x + y\), in units, not the value of \(z\), in units.

The correct answer is \(27\). It's given that a triangular prism has a volume of \(216 cubic centimeters(cm cubed)\) and the volume of a triangular prism is equal to \(B h\), where \(B\) is the area of the base and \(h\) is the height of the prism. Therefore, \(216 = B h\). It's also given that the triangular prism has a height of \(8 cm\). Therefore, \(h = 8\). Substituting \(8\) for \(h\) in the equation \(216 = B h\) yields \(216 = B(8)\). Dividing both sides of this equation by \(8\) yields \(27 = B\). Therefore, the area, \(in cm ^2\), of the base of the prism is \(27\).

Choice C is correct. The perimeter of a quadrilateral is the sum of the lengths of its four sides. It's given that the rectangle has a length of \(13\) and a width of \(6\). It follows that the rectangle has two sides with length \(13\) and two sides with length \(6\). Therefore, the perimeter of the rectangle is \(13 + 13 + 6 + 6\), or \(38\). Choice A is incorrect. This is the sum of the lengths of the two sides with length \(6\), not the sum of the lengths of all four sides of the rectangle. Choice B is incorrect. This is the sum of the lengths of the two sides with length \(13\), not the sum of the lengths of all four sides of the rectangle. Choice D is incorrect. This is the perimeter of a rectangle that has four sides with length \(13\), not two sides with length \(13\) and two sides with length \(6\).

The correct answer is \(4.41\). The area, \(A\), of a circle is given by the formula \(A =π r ^2\), where \(r\) is the radius of the circle. It's given that the area of the circle is \(bπ\) square inches, where \(b\) is a constant, and the radius of the circle is \(2.1\) inches. Substituting \(bπ\) for \(A\) and \(2.1\) for \(r\) in the formula \(A =π r ^2\) yields \(bπ =π(2.1 ^2)\). Dividing both sides of this equation by \(π\) yields \(b = 4.41\). Therefore, the value of \(b\) is \(4.41\).

Choice A is correct. The equation for the volume of a right circular cone is . It’s given that the volume of the right circular cone is cubic feet and the height is 9 feet. Substituting these values for V and h, respectively, gives . Dividing both sides of the equation by gives . Dividing both sides of the equation by 9 gives . Taking the square root of both sides results in two possible values for the radius, or . Since the radius can’t have a -value, that leaves as the only possibility. Applying the quotient property of square roots,, results in, or .Choices B and C are incorrect and may result from incorrectly evaluating . Choice D is incorrect and may result from solving instead of .

Choice B is correct. The perimeter of a triangle is the sum of the lengths of all three of its sides. It's given that the lengths of two sides of a triangle are \(4\) centimeters and \(6\) centimeters. Let \(x\) represent the length, in centimeters, of the third side of this triangle. The sum of the lengths, in centimeters, of all three sides of the triangle can be represented by the expression \(4 + 6 + x\). Since it’s given that the perimeter of the triangle is \(18\) centimeters, it follows that \(4 + 6 + x = 18\), or \(10 + x = 18\). Subtracting \(10\) from both sides of this equation yields \(x = 8\). Therefore, the length, in centimeters, of the third side of this triangle is \(8\). Choice A is incorrect. If the length of the third side of this triangle were \(2\) centimeters, the perimeter, in centimeters, of the triangle would be \(4 + 6 + 2\), or \(12\), not \(18\). Choice C is incorrect. If the length of the third side of this triangle were \(10\) centimeters, the perimeter, in centimeters, of the triangle would be \(4 + 6 + 10\), or \(20\), not \(18\). Choice D is incorrect. If the length of the third side of this triangle were \(24\) centimeters, the perimeter, in centimeters, of the triangle would be \(4 + 6 + 24\), or \(34\), not \(18\).

The correct answer is \(986\). The area, \(A\), of a rectangle is given by \(A = ell w\), where \(ell\) is the length of the rectangle and \(w\) is its width. It’s given that the length of the rectangle is \(34\) centimeters (cm) and the width is \(29\) cm. Substituting \(34\) for \(ell\) and \(29\) for \(w\) in the equation \(A = ell w\) yields \(A =(34)(29)\), or \(A = 986\). Therefore, the area, in square centimeters, of this rectangle is \(986\).

The correct answer is \(2,216\). The surface area of a prism is the sum of the areas of all its faces. A right rectangular prism consists of six rectangular faces, where opposite faces are congruent. It's given that this prism has a length of \(28 cm\), a width of \(15 cm\), and a height of \(16 cm\). Thus, for this prism, there are two faces with area \((28)(15)cm ^2\), two faces with area \((28)(16)cm ^2\), and two faces with area \((15)(16)cm ^2\). Therefore, the surface area, \(in cm ^2\), of the right rectangular prism is \(2(28)(15)+ 2(28)(16)+ 2(15)(16)\), or \(2,216\).

The correct answer is \(518.4\). It's given that the area of the original poster is \(360\) square inches. Let \(ell\) represent the length, in inches, of the original poster, and let \(w\) represent the width, in inches, of the original poster. Since the area of a rectangle is equal to its length times its width, it follows that \(360 = ell w\). It's also given that a copy of the poster is made in which the length and width of the original poster are each increased by \(20 percent sign\). It follows that the length of the copy is the length of the original poster plus \(20 percent sign\) of the length of the original poster, which is equivalent to \(ell + 20/100 ell\) inches. This length can be rewritten as \(ell + 0.2 ell\) inches, or \(1.2 ell\) inches. Similarly, the width of the copy is the width of the original poster plus \(20 percent sign\) of the width of the original poster, which is equivalent to \(w + 20/100 w\) inches. This width can be rewritten as \(w + 0.2 w\) inches, or \(1.2 w\) inches. Since the area of a rectangle is equal to its length times its width, it follows that the area, in square inches, of the copy is equal to \((1.2 ell)(1.2 w)\), which can be rewritten as \((1.2)(1.2)(ell w)\). Since \(360 = ell w\), the area, in square inches, of the copy can be found by substituting \(360\) for \(ell w\) in the expression \((1.2)(1.2)(ell w)\), which yields \((1.2)(1.2)(360)\), or \(518.4\). Therefore, the area of the copy, in square inches, is \(518.4\).

Choice B is correct. Since triangles ABC and DEF are similar and, the length of each side of triangle DEF is two times the length of its corresponding side in triangle ABC. Therefore, the perimeter of triangle DEF is two times the perimeter of triangle ABC. Since the perimeter of triangle ABC is 20, the perimeter of triangle DEF is 40.Choice A is incorrect. This is half, not two times, the perimeter of triangle ABC. Choice C is incorrect. This is two times the perimeter of triangle DEF rather than two times the perimeter of triangle ABC. Choice D is incorrect. This is six times, not two times, the perimeter of triangle ABC.

Choice D is correct. The area of a rectangle is given by \(b h\), where \(b\) is the length of the base of the rectangle and \(h\) is its height. Let \(x\) represent the length, in units, of the base of rectangle \( ABCD\), and let \(y\) represent its height, in units. Substituting \(x\) for \(b\) and \(y\) for \(h\) in the formula \(b h\) yields \(x y\). Therefore, the area, in square units, of \( ABCD\) can be represented by the expression \(x y\). It’s given that the length of each side of \( EFGH\) is \(6\) times the length of the corresponding side of \( ABCD\). Therefore, the length, in units, of the base of \( EFGH\) can be represented by the expression \(6 x\), and its height, in units, can be represented by the expression \(6 y\). Substituting \(6 x\) for \(b\) and \(6 y\) for \(h\) in the formula \(b h\) yields \((6 x)(6 y)\), which is equivalent to \(36 x y\). Therefore, the area, in square units, of \( EFGH\) can be represented by the expression \(36 x y\). It’s given that the area of \( ABCD\) is \(54\) square units. Since \(x y\) represents the area, in square units, of \( ABCD\), substituting \(54\) for \(x y\) in the expression \(36 x y\) yields \(36(54)\), or \(1,944\). Therefore, the area, in square units, of \( EFGH\) is \(1,944\). Choice A is incorrect. This is the area of a rectangle where the length of each side of the rectangle is \(√1 sixth \), not \(6\), times the length of the corresponding side of \( ABCD\). Choice B is incorrect. This is the area of a rectangle where the length of each side of the rectangle is \(√2 thirds \), not \(6\), times the length of the corresponding side of \( ABCD\). Choice C is incorrect. This is the area of a rectangle where the length of each side of the rectangle is \(√6 \), not \(6\), times the length of the corresponding side of \( ABCD\).

Choice D is correct. It's given that rectangle P has an area of \(72\) square inches. If a rectangle with an area of \(20\) square inches is removed from rectangle P, the area, in square inches, of the resulting figure is \(72 -20\), or \(52\). Choice A is incorrect and may result from conceptual or calculation errors. Choice B is incorrect and may result from conceptual or calculation errors. Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is correct. The volume, V, of a right cylinder is given by the formula, where r represents the radius of the base of the cylinder and h represents the height. Since the height is 4 inches longer than the radius, the expression represents the height of each cylindrical container. It follows that the volume of each container is represented by the equation . Distributing the expression into each term in the parentheses yields .Choice A is incorrect and may result from representing the height as instead of . Choice B is incorrect and may result from representing the height as instead of . Choice C is incorrect and may result from representing the volume of a right cylinder as instead of .

Choice D is correct. The area, \(A\), of a circle is given by the formula \(A =π r ^2\), where \(r\) represents the radius of the circle. It’s given that circle \( K\) has a radius of \(4 millimeters(mm)\). Substituting \(4\) for \(r\) in the formula \(A =π r ^2\) yields \(A =π(4)^2\), or \(A = 16π\). Therefore, the area of circle \( K\) is \(16π mm Superscript 2\). It’s given that circle \( L\) has an area of \(100π mm Superscript 2\). Therefore, the total area, \(in mm Superscript 2\), of circles \( K\) and \( L\) is \(16π + 100π\), or \(116π\). Choice A is incorrect. This is the sum of the radii, \(in mm\), of circles \( K\) and \( L\) multiplied by \(π\), not the total area, \(in mm Superscript 2\), of the circles. Choice B is incorrect. This is the sum of the diameters, \(in mm\), of circles \( K\) and \( L\) multiplied by \(π\), not the total area, \(in mm Superscript 2\), of the circles. Choice C is incorrect and may result from conceptual or calculation errors.

The correct answer is \(31\). The circumference of a circle is equal to \(2π r\) centimeters, where \(r\) represents the radius, in centimeters, of the circle, and the diameter of the circle is equal to \(2 r\) centimeters. It's given that a circle has a circumference of \(31π\) centimeters. Therefore, \(31π = 2π r\). Dividing both sides of this equation by \(π\) yields \(31 = 2 r\). Since the diameter of the circle is equal to \(2 r\) centimeters, it follows that the diameter, in centimeters, of the circle is \(31\).

The correct answer is \(66\). It’s given that the right circular cone has a height of \(22\) centimeters \((cm)\) and a base with a diameter of \(6 cm\). Since the diameter of the base of the cone is \(6 cm\), the radius of the base is \(3 cm\). The volume \( V\), \(in cm cubed\), of a right circular cone can be found using the formula \( V = 1 thirdπ r ^2 h\), where \(h\) is the height, \(in cm\), and \(r\) is the radius, \(in cm\), of the base of the cone. Substituting \(22\) for \(h\) and \(3\) for \(r\) in this formula yields \( V = 1 thirdπ(3)^2(22)\), or \( V = 66π\). Therefore, the volume of the cone is \(66π cm cubed\). It’s given that the volume of the cone is \(nπ cm cubed\). Therefore, the value of \(n\) is \(66\).

Choice D is correct. It's given that triangle\( X Y Z\) is similar to triangle \( T U V\). Therefore, each side of triangle \( X Y Z\) is \(k\) times its corresponding side of triangle \( T U V\), where \(k\) is a constant. It follows that the perimeter of triangle \( X Y Z\) is \(k\) times the perimeter of triangle \( T U V\). It's also given that \(line segment T U\) corresponds to \(line segment X Y\) and the length of \(line segment T U\) is \(18\). Let \(x\) represent the length of \(line segment X Y\). It follows that \(x = 18 k\). The table shows that the perimeters of triangles \( T U V\) and \( X Y Z\) are \(37\) and \(333\), respectively. It follows that \(333 = 37 k\), or \(9 = k\). Substituting \(9\) for \(k\) in the equation \(x = 18 k\) yields \(x =(18)(9)\), or \(x = 162\). Therefore, the length of \(line segment X Y\) is \(162\). Choice A is incorrect and may result from conceptual or calculation errors. Choice B is incorrect. This is the length of \(line segment T U\), not the length of \(line segment X Y\). Choice C is incorrect and may result from conceptual or calculation errors.

Choice B is correct. The perimeter of a triangle is the sum of the lengths of all three sides of the triangle. It’s given that the lengths of side \(A B\) and side \(A C\) are \(4\) inches and \(7\) inches, respectively. Let \(x\) represent the length, in inches, of side \(B C\). The sum of the lengths, in inches, of all three sides of triangle \(A B C\) can be represented by the expression \(4 + 7 + x\). Since it’s given that the perimeter of triangle \(A B C\) is \(17\) inches, it follows that \(17 = 4 + 7 + x\), or \(17 = 11 + x\). Subtracting \(11\) from both sides of this equation yields \(6 = x\). Therefore, the length, in inches, of side \(B C\) is \(6\). Choice A is incorrect. This is the length, in inches, of side \(A B\). Choice C is incorrect. This is the length, in inches, of side \(A C\). Choice D is incorrect. This is the sum of the lengths, in inches, of sides \(A B\) and \(A C\).

The correct answer is \(36,504\). The volume of a cube can be found using the formula \( V = s cubed\), where \(s\) represents the edge length of a cube. It’s given that this cube has a volume of \(474,552\) cubic units. Substituting \(474,552\) for \( V\) in \( V = s cubed\) yields \(474,552 = s cubed\). Taking the cube root of both sides of this equation yields \(78 = s\). Thus, the edge length of the cube is \(78\) units. Since each face of a cube is a square, it follows that each face has an edge length of \(78\) units. The area of a square can be found using the formula \(A = s ^2\). Substituting \(78\) for \(s\) in this formula yields \(A = 78 ^2\), or \(A = 6,084\). Therefore, the area of one face of this cube is \(6,084\) square units. Since a cube has \(6\) faces, the surface area, in square units, of this cube is \(6(6,084)\), or \(36,504\).

Choice A is correct. It’s given that the length of each side of a scale model is \(1 tenth\) times the length of the corresponding side of the actual floor of a ballroom. Therefore, the area of the scale model is \((1 tenth)^2\), or \(1/100\), times the area of the actual floor of the ballroom. It’s given that the area of the floor of the ballroom is \(600\) square meters. Therefore, the area, in square meters, of the scale model is \((1/100)(600)\), or \(6\). Choice B is incorrect and may result from conceptual or calculation errors. Choice C is incorrect and may result from conceptual or calculation errors. Choice D is incorrect and may result from conceptual or calculation errors.

Choice D is correct. The volume, \( V\), of a sphere can be found using the formula \( V = 4 thirdsπ r cubed\), where \(r\) is the radius of the sphere. It’s given that the sphere has a radius of \(17/5\) feet. Substituting \(17/5\) for \(r\) in the formula \( V = 4 thirdsπ r cubed\) yields \( V = 4 thirdsπ(17/5)cubed\), which is equivalent to \( V = 4 thirdsπ(4,913/125)\), or \( V = 19,652π/375\). Therefore, the volume, in cubic feet, of the sphere is \(19,652π/375\). Choice A is incorrect and may result from conceptual or calculation errors. Choice B is incorrect. This is the volume, in cubic feet, of a sphere with a radius of \(RootIndex 3 √17/5 \) feet. Choice C is incorrect and may result from conceptual or calculation errors.

Choice A is correct. The area \(A\), in square meters, of a rectangle is the product of its length \(ell\), in meters, and its width \(w\), in meters; thus, \(A = ell w\). It's given that a rectangle has an area of \(63\) square meters and a length of \(9\) meters. Substituting \(63\) for \(A\) and \(9\) for \(ell\) in the equation \(A = ell w\) yields \(63 = 9 w\). Dividing both sides of this equation by \(9\) yields \(7 = w\). Therefore, the width, in meters, of the rectangle is \(7\). Choice B is incorrect. This is the difference between the area, in square meters, and the length, in meters, of the rectangle, not the width, in meters, of the rectangle. Choice C is incorrect. This is the square of the length, in meters, not the width, in meters, of the rectangle. Choice D is incorrect. This is the product of the area, in square meters, and the length, in meters, of the rectangle, not the width, in meters, of the rectangle.

The correct answer is \(1,260\). Since it's given that prisms X and Y are similar, all the linear measurements of prism Y are \(k\) times the respective linear measurements of prism X, where \(k\) is a positive constant. Therefore, the surface area of prism Y is \(k ^2\) times the surface area of prism X and the volume of prism Y is \(k cubed\) times the volume of prism X. It's given that the surface area of prism Y is \(1,450 c m ^2\), and the surface area of prism X is \(58 c m ^2\), which implies that \(1,450 = 58 k ^2\). Dividing both sides of this equation by \(58\) yields \(1,450/58 = k ^2\), or \(k ^2 = 25\). Since \(k\) is a positive constant, \(k = 5\). It's given that the volume of prism Y is \(1,250 c m cubed\). Therefore, the volume of prism X is equal to \(1,250/k cubed c m cubed\), which is equivalent to \(1,250/5 cubed c m cubed\), or \(10 c m cubed\). Thus, the sum of the volumes, in \(c m cubed\), of the two prisms is \(1,250 + 10\), or \(1,260\).

Choice B is correct. If the radius of container A is 16 centimeters and the radius of container B is 25% longer than the radius of container A, then the radius of container B is centimeters. The volume of a cylinder is, where r is the radius of the cylinder and h is its height. Substituting and into yields that the volume of cylinder B is cubic centimeters.Choice A is incorrect and may result from multiplying the radius of cylinder B by the radius of cylinder A rather than squaring the radius of cylinder B. Choice C is incorrect and may result from multiplying the radius of cylinder B by 25 rather than squaring it. Choice D is incorrect and may result from taking the radius of cylinder B to be 25 centimeters rather than 20 centimeters.

Choice C is correct. The base of a cylinder is a circle with a diameter equal to the diameter of the cylinder. The volume, \( V\), of a cylinder can be found by multiplying the area of the circular base, \(A\), by the height of the cylinder, \(h\), or \( V = A h\). The area of a circle can be found using the formula \(A =π r ^2\), where \(r\) is the radius of the circle. It’s given that the diameter of the cylinder is \(8\) inches. Thus, the radius of this circle is \(4\) inches. Therefore, the area of the circular base of the cylinder is \(A =π(4)^2\), or \(16π\) square inches. It’s given that the height \(h\) of the cylinder is \(12\) inches. Substituting \(16π\) for \(A\) and \(12\) for \(h\) in the formula \( V = A h\) gives \( V = 16π(12)\), or \(192π\) cubic inches. Choice A is incorrect. This is the area of the circular base of the cylinder. Choice B is incorrect and may result from using \(8\), instead of \(16\), as the value of \(r ^2\) in the formula for the area of a circle. Choice D is incorrect and may result from using \(8\), instead of \(4\), for the radius of the circular base.

Choice D is correct. The volume, \( V\), of a right square prism is given by the formula \( V = s ^2 h\), where \(s\) represents the length of the edge of the base and \(h\) represents the height of the prism. It’s given that the volume of a right square prism is \(2,880\) cubic units and the length of the edge of the base is \(6\) units. Substituting \(2,880\) for \( V\) and \(6\) for \(s\) in the formula \( V = s ^2 h\) yields \(2,880 =(6 ^2)h\), or \(2,880 = 36 h\). Dividing both sides of this equation by \(36\) yields \(80 = h\). Therefore, the height, in units, of the prism is \(80\). Choice A is incorrect. This is the height, in units, of a right square prism where the length of the edge of the base is \(6\) units and the volume of the prism is \(144 √30 \), not \(2,880\), units. Choice B is incorrect. This is the area, in square units, of the base, not the height, in units, of the prism. Choice C is incorrect. This is the height, in units, of a right square prism where the length of the edge of the base is \(6\) units and the volume of the prism is \(864 √5 \), not \(2,880\), units.

Choice D is correct. The volume, \( V\), of a cube can be found using the formula \( V = s cubed\), where \(s\) is the edge length of the cube. It's given that a cube has an edge length of \(41\) inches. Substituting \(41\) inches for \(s\) in this equation yields \( V = 41 cubed\) cubic inches, or \( V = 68,921\) cubic inches. Therefore, the volume of the cube is \(68,921\) cubic inches. Choice A is incorrect. This is the perimeter, in inches, of the cube. Choice B is incorrect. This is the area, in square inches, of a face of the cube. Choice C is incorrect. This is the surface area, in square inches, of the cube.

The correct answer is \(15/2\). The area, \(A\), of a triangle is given by the formula \(A = 1 half b h\), where \(b\) is the length of the base of the triangle and \(h\) is the height of the triangle. In the right triangle shown, the length of the base of the triangle is \(5\) inches, and the height is \(3\) inches. It follows that \(b = 5\) and \(h = 3\). Substituting \(5\) for \(b\) and \(3\) for \(h\) in the formula \(A = 1 half b h\) yields \(A = 1 half(5)(3)\), which is equivalent to \(A = 1 half(15)\), or \(A = 15/2\). Therefore, the area of the triangle, in square inches, is \(15/2\). Note that 15/2 and 7.5 are examples of ways to enter a correct answer.

The correct answer is \(29\). The volume, \( V\), of a right circular cylinder is given by the formula \( V =π r ^2 h\), where \(r\) is the radius of the base of the cylinder and \(h\) is the height of the cylinder. Since the base of the cylinder is a circle with radius \(r\), the area of the base of the cylinder is \(π r ^2\). It's given that a right circular cylinder has a volume of \(377\) cubic centimeters; therefore, \( V = 377\). It's also given that the area of the base of the cylinder is \(13\) square centimeters; therefore, \(π r ^2 = 13\). Substituting \(377\) for \( V\) and \(13\) for \(π r ^2\) in the formula \( V =π r ^2 h\) yields \(377 = 13 h\). Dividing both sides of this equation by \(13\) yields \(29 = h\). Therefore, the height of the cylinder, in centimeters, is \(29\).

Choice C is correct. The volume of a right circular cylinder is equal to \(π a ^2 b\), where \(a\) is the radius of a base of the cylinder and \(b\) is the height of the cylinder. It’s given that the cylinder shown has a radius of \(r\) and a height of \(h\). It follows that the volume of the cylinder shown is equal to \(π r ^2 h\). It’s given that the second right circular cylinder has a radius of \( R\) and a height of \( H\). It follows that the volume of the second cylinder is equal to \(π R ^2 H\). Choice C gives \( R = 7 r\) and \( H = 8 h\). Substituting \(7 r\) for \( R\) and \(8 h\) for \( H\) in the expression that represents the volume of the second cylinder yields \(π(7 r)^2(8 h)\), or \(π(49 r ^2)(8 h)\), which is equivalent to \(π(392 r ^2 h)\), or \(392(π r ^2 h)\). This expression is equal to \(392\) times the volume of the cylinder shown, \(π r ^2 h\). Therefore, \( R = 7 r\) and \( H = 8 h\) could represent the radius \( R\), in terms of \(r\), and the height \( H\), in terms of \(h\), of the second cylinder. Choice A is incorrect. Substituting \(8 r\) for \( R\) and \(7 h\) for \( H\) in the expression that represents the volume of the second cylinder yields \(π(8 r)^2(7 h)\), or \(π(64 r ^2)(7 h)\), which is equivalent to \(π(448 r ^2 h)\), or \(448(π r ^2 h)\). This expression is equal to \(448\), not \(392\), times the volume of the cylinder shown. Choice B is incorrect. Substituting \(8 r\) for \( R\) and \(49 h\) for \( H\) in the expression that represents the volume of the second cylinder yields \(π(8 r)^2(49 h)\), or \(π(64 r ^2)(49 h)\), which is equivalent to \(π(3,136 r ^2 h)\), or \(3,136(π r ^2 h)\). This expression is equal to \(3,136\), not \(392\), times the volume of the cylinder shown. Choice D is incorrect. Substituting \(49 r\) for \( R\) and \(8 h\) for \( H\) in the expression that represents the volume of the second cylinder yields \(π(49 r)^2(8 h)\), or \(π(2,401 r ^2)(8 h)\), which is equivalent to \(π(19,208 r ^2 h)\), or \(19,208(π r ^2 h)\). This expression is equal to \(19,208\), not \(392\), times the volume of the cylinder shown.

Choice A is correct. The volume of a cube can be found by using the formula \( V = s cubed\), where \( V\) is the volume and \(s\) is the edge length of the cube. Therefore, the volume of the given cube is \( V = 68 cubed\), or \(314,432\) cubic inches. The volume of a sphere can be found by using the formula \( V = 4 thirdsπ r cubed\), where \( V\) is the volume and \(r\) is the radius of the sphere. Therefore, the volume of the given sphere is \( V = 4 thirdsπ(34)cubed\), or approximately \(164,636\) cubic inches. The volume of the space in the cube not taken up by the sphere is the difference between the volume of the cube and volume of the sphere. Subtracting the approximate volume of the sphere from the volume of the cube gives \(314,432 -164,636 = 149,796\) cubic inches. Choice B is incorrect and may result from conceptual or calculation errors. Choice C is incorrect and may result from conceptual or calculation errors. Choice D is incorrect and may result from conceptual or calculation errors.

The correct answer is 360. The volume of a right rectangular prism is calculated by multiplying its dimensions: length, width, and height. Multiplying the values given for these dimensions yields a volume of cubic centimeters.

Choice D is correct. The area, \(A\), of a circle is given by the formula \(A =π r ^2\), where \(r\) is the radius of the circle. It’s given that the circle has a radius of \(43\) meters. Substituting \(43\) for \(r\) in the formula \(A =π r ^2\) yields \(A =π(43)^2\), or \(A = 1,849π\). Therefore, the area, in square meters, of the circle is \(1,849π\). Choice A is incorrect. This is the area, in square meters, of a circle with a radius of \(√43/2 \) meters. Choice B is incorrect. This is the area, in square meters, of a circle with a radius of \(√43 \) meters. Choice C is incorrect. This is the circumference, in meters, of the circle.

Choice A is correct. It's given that the area of a square is \(64\) square inches. The area \(A\), in square inches, of a square is given by the formula \(A = s ^2\), where \(s\) is the side length, in inches, of the square. Substituting \(64\) for \(A\) in this formula yields \(64 = s ^2\). Taking the positive square root of both sides of this equation yields \(8 = s\). Thus, the side length, in inches, of this square is \(8\). Choice B is incorrect and may result from conceptual or calculation errors. Choice C is incorrect. This is the area, in square inches, of the square, not the side length, in inches, of the square. Choice D is incorrect and may result from conceptual or calculation errors.

Choice D is correct. The perimeter, \( P\), of a square can be found using the formula \( P = 4 s\), where \(s\) is the length of each side of the square. It's given that square X has a side length of \(12\) centimeters. Substituting \(12\) for \(s\) in the formula for the perimeter of a square yields \( P = 4(12)\), or \( P = 48\). Therefore, the perimeter of square X is \(48\) centimeters. It’s also given that the perimeter of square Y is \(2\) times the perimeter of square X. Therefore, the perimeter of square Y is \(2(48)\), or \(96\), centimeters. Substituting \(96\) for \( P\) in the formula \( P = 4 s\) gives \(96 = 4 s\). Dividing both sides of this equation by \(4\) gives \(24 = s\). Therefore, the length of one side of square Y is \(24\) centimeters. Choice A is incorrect and may result from conceptual or calculation errors. Choice B is incorrect and may result from conceptual or calculation errors. Choice C is incorrect and may result from conceptual or calculation errors.

The correct answer is \(12\). The volume, \( V\), of a right square prism can be calculated using the formula \( V = s ^2 h\), where \(s\) represents the length of an edge of the base and \(h\) represents the height of the prism. It’s given that the volume of the prism is \(2,016\) cubic units and the height is \(14\) units. Substituting \(2,016\) for \( V\) and \(14\) for \(h\) in the formula \( V = s ^2 h\) yields \(2,016 =(s ^2)(14)\). Dividing both sides of this equation by \(14\) yields \(144 = s ^2\). Taking the square root of both sides of this equation yields two solutions: \(-12 = s\) and \(12 = s\). The length can't be -, so \(12 = s\). Therefore, the length, in units, of an edge of the base is \(12\).

Choice C is correct. The area A of a circle with radius r is given by the formula . Thus, a circle with radius 2 has area, which can be rewritten as . The area of a square with side length s is given by the formula . Thus, if a square has the same area as a circle with radius 2, then . Since the side length of a square must be a positive number, taking the square root of both sides of gives . Using the properties of square roots, can be rewritten as, which is equivalent to . Therefore, .Choice A is incorrect. The side length of the square isn’t equal to the radius of the circle. Choices B and D are incorrect and may result from incorrectly simplifying the expression .

Choice C is correct. The area \(A\), \(in square centimeters(cm ^2)\), of a square with side length \(s\), \(in cm\), is given by the formula \(A = s ^2\). It’s given that the square has a side length of \(55 cm\). Substituting \(55\) for \(s\) in the formula \(A = s ^2\) yields \(A = 55 ^2\), or \(A = 3,025\). Therefore, the area, \(in cm ^2\), of the square is \(3,025\). Choice A is incorrect and may result from conceptual or calculation errors. Choice B is incorrect. This is the perimeter, \(in cm\), of the square, not its area, \(in cm ^2\). Choice D is incorrect and may result from conceptual or calculation errors.

Choice C is correct. The area, \(A\), of a rectangle is given by the formula \(A = ell w\), where \(ell\) represents the length of the rectangle and \(w\) represents its width. It’s given that the rectangle has a length of \(7\) inches and a width of \(6\) inches. Substituting \(7\) for \(ell\) and \(6\) for \(w\) in the formula \(A = ell w\) yields \(A =(7)(6)\), or \(A = 42\). Thus, the area, in square inches, of the rectangle is \(42\). Choice A is incorrect. This is the sum, not the product, of the length and width of the rectangle. Choice B is incorrect and may result from conceptual or calculation errors. Choice D is incorrect. This is twice the area, in square inches, of the rectangle.

The correct answer is \(27,556\). The area of a square is \(s ^2\), where \(s\) is the side length of the square. Let \(x\) represent the length of each side of square B. Substituting \(x\) for \(s\) in \(s ^2\) yields \(x ^2\). It follows that the area of square B is \(x ^2\). It’s given that square A has side lengths that are \(166\) times the side lengths of square B. Since \(x\) represents the length of each side of square B, the length of each side of square A can be represented by the expression \(166 x\). It follows that the area of square A is \((166 x)^2\), or \(27,556 x ^2\). It’s given that the area of square A is \(k\) times the area of square B. Since the area of square A is equal to \(27,556 x ^2\), and the area of square B is equal to \(x ^2\), an equation representing the given statement is \(27,556 x ^2 = k x ^2\). Since \(x\) represents the length of each side of square B, the value of \(x\) must be positive. Therefore, the value of \(x ^2\) is also positive, so it does not equal \(0\). Dividing by \(x ^2\) on both sides of the equation \(27,556 x ^2 = k x ^2\) yields \(27,556 = k\). Therefore, the value of \(k\) is \(27,556\).

Choice A is correct. The volume of a right circular cylinder with a radius of r is the product of the area of the base,, and the height, h. The volume of the right circular cylinder described is and its height is 5. If the radius is r, it follows that . Dividing both sides of this equation by yields . Taking the square root of both sides yields or . Since r represents the radius, the value must be positive. Therefore, the radius is 3.Choice B is incorrect and may result from finding that the square of the radius is 9, but then from dividing 9 by 2, rather than taking the square root of 9. Choice C is incorrect. This represents the square of the radius. Choice D is incorrect and may result from solving the equation for, not r, by dividing by on both sides and then by subtracting, not dividing, 5 from both sides.

The correct answer is \(4,205\). The exterior surface area of a figure is the sum of the areas of all its faces. It's given that the box does not have a lid and that each side of the box is in the shape of a square. Therefore, the box consists of \(5\) congruent square faces. It's also given that the length of each edge is \(29\) inches. Let \(s\) represent the length of an edge of a square. It follows that the area of a square is equal to \(s ^2\). Therefore, the area of each of the \(5\) square faces is equal to \(29 ^2\), or \(841\), square inches. Since the box consists of \(5\) congruent square faces, it follows that the sum of the areas of all its faces, or the exterior surface area of this box without a lid, is \(5(841)\), or \(4,205\), square inches.