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

Dividing the equation by 36, completing the square, and simplifying gives n = 12.

Since \( f(3) = 1024 \), substitute into \( f(x) = 8 \cdot d^x \): \( 8 \cdot d^3 = 1024 \Rightarrow d^3 = 128 \Rightarrow d = 4 \). Now find \( f(-2) \): \( f(-2) = 8 \cdot 4^{-2} = 8 \cdot \frac{1}{16} = 0.5 \).

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 percentage can be calculated using the formula: \( \frac{120}{400} \times 100 = 30\% \).

Calculate the slope with the points provided, then derive the equation of the line. Set \(x = r - 2\) to solve for \(b = 32\).

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

The function \(q(x) = 1.3x\) represents a proportional increase, showing a steady, linear growth with respect to \(x\). Therefore, 'Increasing Linear' is correct.

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 \( g(x) = 30(1.20)^x \).

To find \( f(0) \), substitute \( 0 \) for \( x \): \( f(0) = 250(0.4)^0 \). Since \( (0.4)^0 = 1 \), the expression becomes \( 250 imes 1 = 250 \).

Start by setting up the equation: \( 32,000 = 4,000(2)^{3r} \). Divide both sides by 4,000 to obtain \( 8 = (2)^{3r} \). Taking the logarithm base 2 of both sides yields \( 3 = 3r \). Therefore, \( r = 1 \).

The function \(g(x) = 1.5x\) represents a proportional increase with respect to \(x\). As \(x\) increases, \(g(x)\) also increases linearly. Therefore, the correct answer is 'Increasing Linear.'

Since \( m(3) = 4096 \), substitute into \( m(x) = 16 \cdot q^x \): \( 16 \cdot q^3 = 4096 \Rightarrow q^3 = 256 \Rightarrow q = 4 \). Now find \( m(\frac{1}{4}) \): \( m(\frac{1}{4}) = 16 \cdot 4^{\frac{1}{4}} = 16 \cdot \sqrt[4]{4} = 16 \cdot \sqrt{2} = 16 \sqrt{2} \).

Using the points, calculate the slope and equation of the line, then substitute \(x = q + 5\) to find \(b = -55\).

The percentage can be calculated using the formula: \( \frac{250}{500} \times 100 = 50\% \).

We use the Pythagorean theorem: \( c^2 = 8^2 + 15^2 = 64 + 225 = 289 \). Therefore, \( c = \sqrt{289} = 17 \). The hypotenuse can be written as \( \sqrt{289} \), so \( d = 289 \).

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

Substitute \( (2, 2) \) into both inequalities. Both inequalities are satisfied.

After testing the answer choices, only \( (1, 0) \) satisfies both equations.

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 \( \frac{9}{5} \times 5.00 = 9.00 \).

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 \( p(x) = 50(0.40)^x \).

Factor out the greatest common factor, \(2x\), from the expression: \(2x(x^2 + 9x + 16)\). Then factor \(x^2 + 9x + 16\) as \((x + 4)(x + 9)\). The smallest possible value of \(b\) is 4.

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 the surface area of the cube, we need to determine its side length. The formula for the volume of a cube is \( V = s^3 \). Given \( V = 474552 \), solve for \( s \): \( s = \sqrt[3]{474552} \approx 78 \). The formula for surface area is \( 6s^2 \). Thus, the surface area is \( 6(78)^2 = 36,504 \) square units.

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.

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

Using the point-slope form \( y - y_1 = m(x - x_1) \) with the slope \( m = \frac{3}{5} \) and point \( (2, -4) \), the resulting equation is \( y = \frac{3}{5}x - \frac{22}{5} \).

Convert the area to square miles by dividing by \(1760^2\): \(1,549,000 / 3,097,600 = 0.5\) square miles.

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.

Using the slope \( \frac{1}{3} \) and point \( (-3, 5) \) in the point-slope formula, the resulting equation is \( y = \frac{1}{3}x + 6 \).

Since the function decreases by 45%, it retains 55% of its value at each step. Thus, the multiplier for the decay is 0.55. Therefore, the correct equation is \( q(x) = 14(0.55)^x \).

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 \(q = 40\).

Using the quadratic formula, \(x = \frac{6 \pm \sqrt{36 - 12}}{2} = 3 \pm \sqrt{6}\). So, the solution can be written as \(3 + \sqrt{6}\) and the correct value of \(k\) is 6.

The minimum value of \( y \) occurs at \( x = \frac{-b}{2a} \), giving \( x = \frac{6}{2} = 3 \).

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

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.

Data set A is skewed toward lower values, while data set B is skewed toward higher values. Therefore, the mean of Data set A is less than the mean of Data set B.

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.

Rearrange and substitute to find that \( y = 10 \).

To find the area in square miles, divide by \(1760^2\): \(23,184,000 / 3,097,600 = 7.48\) square miles.

The leakage rate is 5 gallons per hour. Over one day (24 hours), the volume leaked is 5 * 24 = 120 gallons. Converting to pints (1 gallon = 8 pints), the answer is 960 pints.

The function \(h(x) = 0.8x\) represents a proportional decrease with respect to \(x\), where \(h(x)\) decreases as \(x\) increases. Therefore, 'Decreasing Linear' is the correct answer.

Calculate percentages: Candidate P received \( \frac{725}{1000} = 0.725 \), or 72.5%. Candidate Q received \( \frac{275}{1000} = 0.275 \), or 27.5%. If 8,000 people vote, Candidate P would receive \( 0.725 \times 8000 = 5800 \) votes, and Candidate Q \( 0.275 \times 8000 = 2200 \) votes. The difference is \( 5800 - 2200 = 3600 \) votes.

Since \( j(5) = 15625 \), substitute into \( j(x) = 50 \cdot k^x \): \( 50 \cdot k^5 = 15625 \Rightarrow k^5 = 312.5 \Rightarrow k = 5 \). Now find \( j(0.5) \): \( j(0.5) = 50 \cdot 5^{0.5} = 50 \cdot \sqrt{5} = 50 \sqrt{5} \).

The system has no solution when the slopes are the same but the lines have different intercepts. Solving this condition gives n = 45.