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

The vertex of the function \( f(x) = \frac{1}{6}(x - 10)^2 + 4 \) occurs at \( (10, 4) \). This represents the minimum height of the basketball. The vertex form of a quadratic function, \( a(x - h)^2 + k \), indicates the vertex as \( (h, k) \). Since \( h = 10 \) and \( k = 4 \), the minimum height is 4 feet above the court.

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

In the function h(x) = -2x + 20, the y-intercept (where x = 0) represents the initial amount of water the gardener had before filling any watering cans. When x = 0, h(0) = 20, indicating that the gardener started with 20 liters of water.

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 people who voted in favor of the policy be x. According to the problem, 5 times as many people voted against it, so the number against is 5x. The article states that 40,000 more people voted against it, so the equation is 5x = x + 40,000. Solving for x gives 10,000.

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.

Using elimination or substitution, we solve for \( y = 5 \).

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

The sum of the solutions of a polynomial equation can be determined using the relationship between the coefficients and the sum of the roots. For each factor, find the solutions and sum them. The given information specifies that the total sum of solutions is \( \frac{20}{3} \), which leads to solving for \( p \).

Dividing the equation by 9, completing the square, and simplifying gives n = 6.

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.

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.

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, \( r = \frac{1}{3} \).

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

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 \(x = 8\).

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

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

The decay factor is 0.75 per half-year, which means it decays by 25% every six months. Over a year, this represents a 25% drop, so \(q = 25\).

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 \( a + b = 3 - 15 = -12 \).

The population doubles every 30 minutes, so it doubles twice per hour. After \( x \) hours, the population grows by a factor of \( 2^{2x} \).

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

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 \( f(-2) \): \( f(-2) = 8 \cdot 4^{-2} = 8 \cdot \frac{1}{16} = 0.5 \).

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

First, find the measure of angle B in radians: \( \frac{\pi}{4} + \frac{3\pi}{8} = \frac{5\pi}{8} \). Then, convert to degrees by multiplying by \( \frac{180}{\pi} \): \( \frac{5\pi}{8} \cdot \frac{180}{\pi} = 112.5 \) degrees.

Since the decay factor is 0.82 per quarter, this gives a 18% decrease each quarter. Calculating this for a year results in approximately a 51% annual decay rate, so \(r = 51\).

Expanding the left side gives \( 5x + 15 \), which matches the right side. Therefore, the equation is true for all values of x, leading to infinitely many solutions.

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.'

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.

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.

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

Calculate the percentages: Candidate A received \( \frac{600}{1000} = 0.6 \), or 60%. Candidate B received \( \frac{400}{1000} = 0.4 \), or 40%. If 10,000 people vote, Candidate A is expected to receive \( 0.6 \times 10000 = 6000 \) votes, and Candidate B \( 0.4 \times 10000 = 4000 \) votes. The difference is \( 6000 - 4000 = 2000 \) votes.

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, \( r = \frac{1}{2} \).

Calculate percentages: Candidate X received \( \frac{350}{800} = 0.4375 \), or 43.75%. Candidate Y received \( \frac{450}{800} = 0.5625 \), or 56.25%. If 5,600 people vote, Candidate X would receive \( 0.4375 \times 5600 = 2450 \) votes, and Candidate Y \( 0.5625 \times 5600 = 3150 \) votes. The difference is \( 3150 - 2450 = 700 \) votes.

In the function k(x) = -8x + 64, the y-intercept (where x = 0) represents the initial amount of concrete the worker had before building any foundations. When x = 0, k(0) = 64, indicating that the worker started with 64 cubic feet of concrete.

Expanding the equation, we obtain \( 2x^2 - 3x + 15 = 15x - 5x^2 \). Rearranging terms, we get \( 7x^2 - 18x + 15 = 0 \). Using the quadratic formula, the sum of the solutions is given by \( -b/a \), where \( a = 7 \) and \( b = -18 \), which yields \( 18/7 \).

Since the function decreases by 30%, it retains 70% of its value at each step. Thus, the multiplier for the decay is 0.70. Therefore, the correct equation is \( f(x) = 100(0.70)^x \).

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

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

With \( h(4) = 810 \), substitute into \( h(x) = 10 \cdot c^x \): \( 10 \cdot c^4 = 810 \Rightarrow c^4 = 81 \Rightarrow c = 3 \). Now find \( h(-1) \): \( h(-1) = 10 \cdot 3^{-1} = 10 \cdot \frac{1}{3} = \frac{10}{3} \).

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

Dividing the equation by 16, completing the square, and simplifying gives n = 8.

In similar triangles, corresponding angles have the same trigonometric values. Thus, \( \sin(J) = \sin(G) = \frac{24}{25} \).

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.