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

To find \( p \): \( 50 = \frac{p}{100} \times 80 \implies p = \frac{50 \times 100}{80} = 62.5 \). Calculate \( 62.5 \% \) of \( 50 \): \( \frac{62.5}{100} \times 50 = 31.25 \).

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 \( 20/8 = 5/2 \).

Let the number of small candles be represented by \( x \), and the number of large candles by \( y \). The total number of candles is at least 200, so we have the inequality:

\( x + y geq 200 \).

The budget constraint is that the total cost must not exceed $2,200. Therefore, we have the equation:

\( 4.90x + 11.60y leq 2,200 \).

We want to maximize the number of large candles, \( y \). First, solve for \( x \) in terms of \( y \) using the inequality:

\( x = 200 - y \).

Substitute this into the budget constraint:

\( 4.90(200 - y) + 11.60y leq 2,200 \)

Simplify:

\( 980 - 4.90y + 11.60y leq 2,200 \)

Combine like terms:

\( 980 + 6.70y leq 2,200 \)

Subtract 980 from both sides:

\( 6.70y leq 1,220 \)

Solve for \( y \):

\( y leq rac{1,220}{6.70} approx 182.09 \)

Since the number of candles must be an integer, the maximum number of large candles is 182.

Since the function decreases by 60%, it retains 40% of its value at each step. Thus, the multiplier for the decay is 0.40. Therefore, the correct equation is \( p(x) = 50(0.40)^x \).

To cover the walls twice, double the total area \(w\) to get \(2w\). Since each gallon covers 200 square feet, divide \(2w\) by 200 to get \(P = \frac{w}{100}\).

The population triples every hour, so the growth factor per hour is 3. Since there are 24 hours in a day, the population after \( x \) days is \( p = 1,000 \cdot 3^{24x} \).

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.

Substitute \( b = 8 \) into the equation: \( 2.5d + 3.5(8) = 70 \Rightarrow 2.5d + 28 = 70 \Rightarrow 2.5d = 42 \Rightarrow d = 16 \).

The percentage can be calculated using the formula: \( \frac{62.5}{250} \times 100 = 25\% \).

Rewrite \(f(x)\) in vertex form as \(-3(x - 2)^2 + 7\). The maximum of \(f(x)\) is at \(x = 2\), so \(g(x) = f(x - 6)\) shifts this maximum to \(x = 8\).

For a quadratic equation, the minimum value occurs at \( x = \frac{-b}{2a} \). Here, \( x = \frac{16}{2} = 8 \).

The values and frequencies are symmetric across the two data sets, which means the mean values are the same.

Using the quadratic formula, \(x = \frac{10 \pm \sqrt{100 - 8}}{2} = 5 \pm \sqrt{23}\). So, the solution can be written as \(5 + \sqrt{23}\) and the correct value of \(k\) is 23.

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

Both equations are identical, which means the lines coincide perfectly and intersect at infinitely many points.

To find the surface area, find the side length first. The volume of the cube is \( V = s^3 \). Given \( V = 729 \), \( s = \sqrt[3]{729} = 9 \). The surface area formula is \( 6s^2 \). Therefore, the surface area is \( 6(9)^2 = 486 \) square units.

The two lines are parallel because their slopes are equal. Dividing the first equation by 2 yields \( 3.5x + y = 7.5 \). Since this differs from the second equation, the lines are parallel and do not intersect.

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 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 \( \frac{9}{5} \times 1.50 = 2.70 \).

Expanding the equation, we obtain \( x^2 + 2x - 10 = 3x^2 - 15x \). Rearranging terms, we get \( -2x^2 + 17x - 10 = 0 \). Using the quadratic formula, the sum of the solutions is given by \( -b/a \), where \( a = -2 \) and \( b = 17 \), which yields \( 17/2 \).

Convert the area to square miles by dividing by \(1760^2\): \(10,240,000 / 3,097,600 = 3.31\) 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 \( a + b = 3 - 15 = -12 \).

Plug in the slope \( m = -2 \) and point \( (3, 7) \) into the point-slope formula to find the equation \( y = -2x + 13 \).

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 \( \frac{9}{5} \times 3.00 = 5.40 \).

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

To find the doubling time, set \( h(t) = 2 \cdot h(0) \) and solve for \( t \): \( 45,000 \cdot (1.04)^{t/200} = 90,000 \). Use logarithms to solve for \( t \) and convert to hours.

To cover the walls twice, double the total area \(w\) to get \(2w\). Since each gallon covers 300 square feet, divide \(2w\) by 300 to get \(P = \frac{w}{150}\).

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, \( a + b = 5 + 10 = 15 \).

Let the number of voters who voted in favor of the bill be w. According to the problem, 3 times as many voters voted against, so the number against is 3w. The report states that 18,000 more people voted against, so the equation is 3w = w + 18,000. Solving for w gives 9,000, and the number against is 3(9,000) = 27,000.

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

Since both sides of the equation are identical, the equation is true for all values of x. Therefore, there are infinitely many solutions.

To have no real solution, the discriminant of \(x(m x - 40) + 25 = 0\) must be negative. Solving gives \(\)m^2 < 25[/latex], so the smallest integer for [latex]m[/latex] is 5.

Substitute \( (-14, 0) \) into both inequalities. Both inequalities are satisfied.

First, find the measure of angle Y in radians: \( \frac{\pi}{6} + \frac{\pi}{3} = \frac{\pi}{2} \). Then, convert to degrees by multiplying by \( \frac{180}{\pi} \): \( \frac{\pi}{2} \cdot \frac{180}{\pi} = 90 \) degrees.

Since the triangles are similar, corresponding angles have the same trigonometric ratios. Thus, \( \sin(E) = \sin(B) = \frac{9}{15} \).

The half-life is 3 years, so the remaining quantity after \( x \) minutes is \( Q = 5 \cdot \left(\frac{1}{2}\right)^{\frac{x}{3 \cdot 365 \cdot 24 \cdot 60}} \).

The lines have different slopes, indicating that they will intersect at exactly one point.

Since triangles \( \triangle ABC \) and \( \triangle DEF \) are congruent, angle \( C \) corresponds to angle \( F \). Therefore, angle \( F \) must be 55°.

Since the side lengths of square M are 15 times those of square N, the area of square M is \( 15^2 = 225 \) times the area of square N. Therefore, \( k = 225 \).

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 \(m = 56\).

Substitute \( (1, 5) \) into both inequalities. Both inequalities are satisfied.

Start by setting up the equation: \( 24,000 = 3,000(2)^{8r} \). Divide both sides by 3,000 to obtain \( 8 = (2)^{8r} \). Taking the logarithm base 2 of both sides gives \( 3 = 8r \). Therefore, \( r = \frac{3}{8} \).

In the equation 5𝑏 + 40𝑐 = 2,450, the coefficient 5 corresponds to the sports field, meaning 𝑏 represents the number of benches per acre in the sports field.

To find the surface area, first find the side length using \( V = s^3 \). Given \( V = 27000 \), find \( s = \sqrt[3]{27000} = 30 \). The surface area is given by \( 6s^2 \). Therefore, the surface area is \( 6(30)^2 = 5400 \) square units.