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

Since \( g(2) = 625 \), substitute into \( g(x) = 25 \cdot b^x \) to solve for \( b \): \( 25 \cdot b^2 = 625 \Rightarrow b^2 = 25 \Rightarrow b = 5 \). Now find \( g(\frac{1}{2}) \): \( g(\frac{1}{2}) = 25 \cdot 5^{\frac{1}{2}} = 25 \cdot \sqrt{5} = 25 \sqrt{5} \).

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

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 \( m(\frac{1}{4}) \): \( m(\frac{1}{4}) = 16 \cdot 4^{\frac{1}{4}} = 16 \cdot \sqrt[4]{4} = 16 \cdot \sqrt{2} = 16 \sqrt{2} \).

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 \(q = 40\).

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

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

To find \( a \), set \( p(x) = 0 \): \( 3x + 9 = 0 Rightarrow x = -3 \). Thus, \( a = -3 \). For \( b \), evaluate \( p(0) = 9 \). Therefore, \( a + b = -3 + 9 = 6 \).

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 \( q(x) = 14(0.55)^x \).

We use the Pythagorean theorem: \( c^2 = 7^2 + 24^2 = 49 + 576 = 625 \). Therefore, \( c = \sqrt{625} = 25 \). Since the length is already in the form \( \sqrt{625} [/latex}, the value of [latex] d \) is 625.

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

Factor out the greatest common factor, \(4x\), from the expression: \(4x(x^2 + 8x + 24)\). Then factor \(x^2 + 8x + 24\) as \((x + 4)(x + 6)\). The smallest possible value of \(b\) is 4.

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.

First, find the measure of angle Q in radians: \( \frac{5\pi}{6} + \frac{\pi}{4} = \frac{14\pi}{12} = \frac{7\pi}{6} \). Then, convert to degrees by multiplying by \( \frac{180}{\pi} \): \( \frac{7\pi}{6} \cdot \frac{180}{\pi} = 210 \) degrees.

To have no real solution, the discriminant of \(x(r x - 120) + 64 = 0\) must be negative. Solving gives \(\)r^2 < 64[/latex], so the smallest integer for [latex]r[/latex] is 8.

The percentage can be calculated using the formula: \( \frac{350}{700} \times 100 = 50\% \).

Dividing the equation by 4, completing the square, and simplifying gives n = 4.

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

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

Substitute \( s = 10 \) into the equation: \( 4p + 2(10) = 100 \Rightarrow 4p + 20 = 100 \Rightarrow 4p = 80 \Rightarrow p = 20 \).

Since \(g(x) = f(x+4) = 7(2)^{x+4} = 7 \cdot 16 \cdot (2)^x = 112(2)^x\), the correct answer is g(x) = 112(2)^x.

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 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.

Rewrite \(f(x)\) as \((x - 5)^2\). The minimum of \(f(x)\) is at \(x = 5\), so \(g(x) = f(x - 2)\) shifts this minimum to \(x = 7\).

To ensure no solution, we set the slopes equal, indicating b = 48 based on intercept calculations.

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.

First, find the measure of angle N in radians: \( \frac{3\pi}{4} + \frac{\pi}{6} = \frac{11\pi}{12} \). Then, convert to degrees by multiplying by \( \frac{180}{\pi} \): \( \frac{11\pi}{12} \cdot \frac{180}{\pi} = 165 \) degrees.

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.

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.

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.

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

First, find the measure of angle T in radians: \( \frac{2\pi}{3} + \frac{\pi}{4} = \frac{11\pi}{12} \). Then, convert to degrees by multiplying by \( \frac{180}{\pi} \): \( \frac{11\pi}{12} \cdot \frac{180}{\pi} = 165 \) degrees.

To have no real solution, the discriminant of \(x(p x - 90) + 49 = 0\) must be negative. Solving gives \(\)p^2 < 49[/latex], so the smallest integer for [latex]p[/latex] is 7.

The mean of each data set is centered around 15, as both data sets have symmetric distributions around this value.

Since the side lengths of square C are 8 times those of square D, the area of square C is \( 8^2 = 64 \) times the area of square D. Therefore, \( k = 64 \).

Since the decay factor is 0.70 annually, this translates to a 30% decrease per year, so \(q = 30\).

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 investment grows by 8% per month, so the monthly growth factor is 1.08. After \( x \) years, the equation is \( V = 10,000 \cdot 1.08^{12x} \).

To find the doubling time, solve \( j(t) = 2 \cdot j(0) \), leading to \( 100,000 \cdot (1.03)^{t/350} = 200,000 \). Taking logarithms of both sides allows solving for \( t \) and then converting to hours.

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.

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.

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

Since the function increases by 50%, it becomes 1.50 times its previous value at each step. Thus, the multiplier for growth is 1.50. Therefore, the correct equation is \( k(x) = 80(1.50)^x \).

Both sides simplify to \( 8z + 4 \), making the equation true for all values of z. Therefore, there are infinitely many solutions.

To find the doubling time, set \( b(t) = 2 \cdot b(0) \) and solve for \( t \): \( 15,000 \cdot (1.06)^{t/100} = 30,000 \). Use logarithms to isolate \( t \) and convert to hours.