SAT Randomized Questions - 1 Full Math Practice Test - Answers and Detailed Explanations at the END

To find \( a \), set \( g(x) = 0 \) and solve: \( -2x + 10 = 0 Rightarrow x = 5 \), so \( a = 5 \). For the y-intercept \( b \), set \( x = 0 \): \( g(0) = 10 \). Therefore, \( a + b = 5 + 10 = 15 \).

Given \(\frac{m}{n} = 3\), rewrite as \(m = 3n\). Substitute into \(\frac{18m}{pn} = 3\) to get \(\frac{18(3n)}{pn} = 3\). Simplify to \(\frac{54n}{pn} = 3\), then cancel \(n\) to find \(p = 18\).

Expanding the equation, we obtain \( 2x^2 - 8x + 12 = 12x - 6x^2 \). Rearranging terms, we get \( 8x^2 - 20x + 12 = 0 \). Using the quadratic formula, the sum of the solutions is given by \( -b/a \), where \( a = 8 \) and \( b = -20 \), which yields \( 20/8 = 5/2 \).

The two equations are not multiples of each other, but after simplifying, they end up being parallel lines. The slopes are the same but the y-intercepts are different, which means the lines never intersect.

Substitute \( (-1, 3) \) into both inequalities. Both inequalities are satisfied.

The two equations are actually multiples of each other. Dividing the second equation by 2 yields \( 3x + 5y = 25 \). Thus, both lines coincide, which means they intersect at infinitely many points.

Solving for \( y \) using substitution or elimination, we find that \( y = 30 \).

To find the area in square miles, divide by \(1760^2\): \(8,673,280 / 3,097,600 = 2.8\) square miles.

The rate of leakage is 7 milliliters per minute. Over one hour (60 minutes), the volume leaked is 7 * 60 = 420 milliliters. Converting to liters (1 liter = 1000 mL), the answer is 0.42 liters.

From \(\frac{x}{y} = 8\), we know \(x = 8y\). Substitute into \(\frac{56x}{my} = 8\) to get \(\frac{56(8y)}{my} = 8\). Simplify to \(\frac{448y}{my} = 8\), cancel \(y\) and solve to find \(m = 56\).

Start by setting up the equation with the given values: \( 64,000 = 8,000(2)^{6r} \). Divide both sides by 8,000 to get \( 8 = (2)^{6r} \). Taking the logarithm base 2 of both sides gives \( 3 = 6r \). Therefore, \( r = \frac{1}{2} \).

The percentage can be calculated using the formula: \( \frac{\text{Part}}{\text{Whole}} \times 100 \). Here, \( \frac{50}{200} \times 100 = 25\% \).

The vertex of the function \( f(x) = \frac{1}{9}(x - 7)^2 + 3 \) occurs at \( (7, 3) \). This represents the minimum height of the metal ball. The vertex form of a quadratic function, \( a(x - h)^2 + k \), indicates the vertex as \( (h, k) \). Since \( h = 7 \) and \( k = 3 \), the minimum height is 3 inches above the ground.

The lines have the same slope but different intercepts, indicating that they are parallel and do not intersect.

Since \( \triangle PQR \) and \( \triangle STU \) are congruent, corresponding angles are equal. Angle \( Q \) and \( T \) are right angles, so each measures 90°. Therefore, angle \( U \) must be equal to angle \( P \), which is 25°.

To find \( p \): \( 75 = \frac{p}{100} \times 150 \implies p = \frac{75 \times 100}{150} = 50 \). Calculate \( 50 \% \) of \( 75 \): \( \frac{50}{100} \times 75 = 37.5 \).

To find \( p \): \( 45 = \frac{p}{100} \times 90 \implies p = \frac{45 \times 100}{90} = 50 \). Calculate \( 50 \% \) of \( 45 \): \( \frac{50}{100} \times 45 = 22.5 \).

Since the triangles are similar, corresponding angles have equal trigonometric ratios, so \( \sin(U) = \sin(R) = \frac{40}{41} \).

Isolate \(10a + 3b\) by multiplying both sides by \((10a + 3b)\), resulting in \(m = q(10a + 3b)\). Dividing by \(q\) gives \(10a + 3b = \frac{m}{q}\).

Using \( x = \frac{-b}{2a} \), we find the minimum value occurs at \( x = \frac{12}{2} = 6 \).

In the equation 4𝑛 + 25𝑚 = 1,150, the coefficient 4 corresponds to the bird sanctuary, meaning 𝑛 represents the number of nests per hectare in the bird sanctuary.

To convert from square yards to square miles, divide the area by \(1760^2\): \(5,904,900 / 3,097,600 = 1.91\) square miles.

To find \( a \), set \( f(x) = 0 \) and solve for \( x \): \( 5x - 15 = 0 Rightarrow x = 3 \). The x-intercept is \( (3, 0) \), so \( a = 3 \). The y-intercept \( b \) is found by setting \( x = 0 \): \( f(0) = -15 \), so \( b = -15 \). Then \( a + b = 3 - 15 = -12 \).

In the equation 10𝑎 + 30𝑏 = 1,800, the coefficient 10 corresponds to the orchard, meaning 𝑎 represents the number of apple trees per acre in the orchard.

Set up the equation \( 24,000 = 1,500(2)^{12r} \). Divide both sides by 1,500 to obtain \( 16 = (2)^{12r} \). Taking the logarithm base 2 of both sides yields \( 4 = 12r \). Therefore, \( r = \frac{1}{3} \).

Using \( P = 40,000 \) and \( C = 5,000 \), set up the equation \( 40,000 = 5,000(2)^{5r} \). Dividing both sides by 5,000 gives \( 8 = (2)^{5r} \). Taking the logarithm base 2 of both sides yields \( 3 = 5r \). Therefore, \( r = \frac{3}{5} \).

To find the doubling time, set \( g(t) = 2 \cdot g(0) \). Thus, \( 20,000 \cdot (1.05)^{t/300} = 40,000 \). Taking logarithms, isolate \( t \) and convert the answer to hours.

In the function g(x) = -3x + 50, the y-intercept (where x = 0) represents the initial amount of dough the baker had before making any loaves. When x = 0, g(0) = 50, indicating that the baker started with 50 pounds of dough.

With \(p(x) = 0.75x\), the function decreases linearly as \(x\) increases, representing a consistent reduction in value. Hence, the answer is 'Decreasing Linear.'

Calculate percentages: Candidate L received \( \frac{600}{1000} = 0.6 \), or 60%. Candidate M received \( \frac{400}{1000} = 0.4 \), or 40%. If 7,500 people vote, Candidate L would receive \( 0.6 \times 7500 = 4500 \) votes, and Candidate M \( 0.4 \times 7500 = 3000 \) votes. The difference is \( 4500 - 3000 = 1500 \) votes.

Since \(\frac{c}{d} = 2\), we have \(c = 2d\). Substitute into \(\frac{40c}{qd} = 2\) to get \(\frac{40(2d)}{qd} = 2\). Simplify to \(\frac{80d}{qd} = 2\), cancel \(d\) and solve to find \(q = 40\).

Congruent triangles have corresponding angles that are equal. Since \( L \) and \( O \) correspond, angle \( O \) must be 40°.

The vertex of the function \( f(x) = \frac{1}{16}(x - 5)^2 + 2 \) occurs at \( (5, 2) \). This represents the minimum height of the roller coaster car. The vertex form of a quadratic function, \( a(x - h)^2 + k \), indicates the vertex as \( (h, k) \). Since \( h = 5 \) and \( k = 2 \), the minimum height is 2 feet above the ground.

The function increases by 20%, which means it becomes 1.20 times its previous value at each step. The multiplier is 1.20. Therefore, the correct equation is \( g(x) = 30(1.20)^x \).

Since \(g(x) = f(x+3) = 3(6)^{x+3} = 3 \cdot 216 \cdot (6)^x = 648(6)^x\), the correct answer is g(x) = 648(6)^x.

We use the Pythagorean theorem: \( c^2 = 5^2 + 12^2 = 25 + 144 = 169 \). Therefore, \( c = \sqrt{169} = 13 \). Since the length is already in the form \( \sqrt{169} [/latex}, the value of [latex] d \) is 169.

Solving by elimination or substitution, we find that \( y = 14 \).

To find \( f(0) \), substitute \( 0 \) for \( x \) in the function: \( f(0) = 300(0.2)^0 \). Since any non-zero number raised to the power of \( 0 \) is \( 1 \), the expression becomes \( 300 imes 1 = 300 \).

Using the quadratic formula, \(x = \frac{8 \pm \sqrt{64 - 28}}{2} = 4 \pm \sqrt{9}\). So, the solution can be written as \(4 + \sqrt{9}\) and the correct value of \(k\) is 9.

To isolate \(3y + 8z\), multiply both sides by \((3y + 8z)\) to obtain \(h = x(3y + 8z)\), then divide by \(x\) to yield \(3y + 8z = \frac{h}{x}\).

For the system to have no solution, the slopes must be equal. Converting both equations to slope-intercept form reveals that k must equal -14.

The change in degrees Fahrenheit is determined by multiplying the temperature increase in kelvins by \( \frac{9}{5} \), since the \( x \) term in the function is being multiplied by this factor. Therefore, the increase in Fahrenheit temperature is \( \frac{9}{5} \times 4.50 = 8.10 \).

To find \( a \), set \( q(x) = 0 \): \( -7x + 21 = 0 Rightarrow x = 3 \), so \( a = 3 \). For \( b \), evaluate \( q(0) = 21 \). Thus, \( a + b = 3 + 21 = 24 \).

For a quadratic in the form \( y = ax^2 + bx + c \), the minimum or maximum value of \( y \) occurs at \( x = \frac{-b}{2a} \). Here, \( a = 1 \) and \( b = -10 \), so \( x = \frac{10}{2} = 5 \).