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

To isolate \(2p + 9q\), multiply both sides by \((2p + 9q)\), resulting in \(g = w(2p + 9q)\). Dividing by \(w\) gives \(2p + 9q = \frac{g}{w}\).

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

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.

Calculate the sum of the solutions for each factor separately. Use the given total sum of 10 to find \( p \).

We use the Pythagorean theorem: \( c^2 = 16^2 + 30^2 = 256 + 900 = 1156 \). Therefore, \( c = \sqrt{1156} = 34 \). The hypotenuse can be written as \( 2\sqrt{289} \), so \( d = 289 \).

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.

Since the triangles are congruent, angle \( Z \) is equal to angle \( C \), so the measure of angle \( C \) is 70°.

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

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

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

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.

The decay factor is 0.90 per three months, equating to a 10% drop per quarter. After one year, this approximates to a 34% annual decay rate, so \(m = 34\).

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 \(q = 25\).

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

Rewrite \(f(x)\) in vertex form as \(5(x - 2)^2 + 75\). The minimum of \(f(x)\) is at \(x = 2\), so \(g(x) = f(x + 3)\) shifts this to \(x = -1\).

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.

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.

To cover the walls twice, double the total area \(w\) to get \(2w\). Since each gallon covers 150 square feet, divide \(2w\) by 150 to get the equation \(P = \frac{w}{75}\).

To have no real solution, the discriminant of \(x(n x - 72) + 36 = 0\) must be negative. Solving gives \(\)n^2 < 36[/latex], so the smallest integer for [latex]n[/latex] is 6.

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.

To isolate \(7m + 5n\), multiply both sides by \((7m + 5n)\) to get \(k = p(7m + 5n)\), then divide by \(p\) to find \(7m + 5n = \frac{k}{p}\).

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 isolate \(5x + 6y\), multiply both sides by \((5x + 6y)\), resulting in \(n = t(5x + 6y)\). Dividing by \(t\) gives \(5x + 6y = \frac{n}{t}\).

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.

Since \(g(x) = f(x+1) = 6(10)^{x+1} = 6 \cdot 10 \cdot (10)^x = 60(10)^x\), the correct answer is g(x) = 60(10)^x.

The volume of a cube is given by \( V = s^3 \). Here, \( V = 64000 \), so \( s = \sqrt[3]{64000} = 40 \). The surface area is \( 6s^2 \). Therefore, the surface area is \( 6(40)^2 = 9600 \) square units.

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 \( k(x) = 80(1.50)^x \).

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.

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

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 \( p \).

To find \( f(0) \), substitute \( 0 \) for \( x \): \( f(0) = 400(0.5)^0 \). Since \( (0.5)^0 = 1 \), the expression becomes \( 400 imes 1 = 400 \).

Calculate the slope using the two points and find the equation of the line. Substitute \(x = m - 4\) to find the y-intercept \(b = 45\).

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.

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.

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 \(P = \frac{w}{100}\).

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 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.

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, \( r = \frac{3}{8} \).

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 \( y = \frac{3}{5}x - \frac{22}{5} \).

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

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

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 \( d = 289 \).

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.

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