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

Factor out the greatest common factor, \(3x\), from the expression: \(3x(x^2 + 9x + 18)\). Then factor \(x^2 + 9x + 18\) as \((x + 3)(x + 6)\). The smallest possible value of \(b\) is 3.

Substitute \( y = 24 \) into the equation: \( 3.5x + 7(24) = 140 \Rightarrow 3.5x + 168 = 140 \Rightarrow 3.5x = -28 \). Since \( x \) cannot be negative, the shelter cannot care for any cats, making this question tricky.

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

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

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 \(x = -1\).

Expanding the right side results in \( 10a - 3 \), which is identical to the left side. Therefore, the equation holds for all values of a, implying infinitely many solutions.

Substitute values from each answer choice into both equations. Only \( (5, 10) \) satisfies both equations.

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.

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.'

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 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 determine the doubling time, solve \( p(t) = 2 \cdot p(0) \), which gives \( 75,000 \cdot (1.02)^{t/250} = 150,000 \). Take the logarithm of both sides, isolate \( t \), and convert to hours.

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

Factor out the greatest common factor, \(5x\), from the expression: \(5x(x^2 + 7x + 12)\). Then factor \(x^2 + 7x + 12\) as \((x + 3)(x + 4)\). The smallest possible value of \(b\) is 3.

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 \(x = 7\).

The vertex of the function \( f(x) = \frac{1}{4}(x - 8)^2 + 6 \) occurs at \( (8, 6) \). This represents the minimum height of the drone. The vertex form of a quadratic function, \( a(x - h)^2 + k \), indicates the vertex as \( (h, k) \). Since \( h = 8 \) and \( k = 6 \), the minimum height is 6 meters above the ground.

Rewrite \(f(x)\) in vertex form by completing the square, yielding \(f(x) = 3(x + 4)^2 - 5\). The minimum of \(f(x)\) is at \(x = -4\), so \(g(x) = f(x - 4)\) shifts this to \(x = 0\).

To find \( a \), set \( r(x) = 0 \): \( 8x - 16 = 0 Rightarrow x = 2 \). Thus, \( a = 2 \). For \( b \), evaluate \( r(0) = -16 \). Therefore, \( a + b = 2 - 16 = -14 \).

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

To find \( p \): \( 120 = \frac{p}{100} \times 200 \implies p = \frac{120 \times 100}{200} = 60 \). Calculate \( 60 \% \) of \( 120 \): \( \frac{60}{100} \times 120 = 72 \).

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

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

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

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

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 \( f(0) \), substitute \( 0 \) for \( x \) in the function: \( f(0) = 300(0.2)^0 \). Since any non-zero number raised to the power of \( 0 \) is \( 1 \), the expression becomes \( 300 imes 1 = 300 \).

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

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

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.

Rewrite both equations in slope-intercept form. To have no solution, the slopes must be equal but the y-intercepts must differ. Solving, we find m = -6.

Let the number of people who voted in favor of the measure be z. According to the problem, 4 times as many people voted against, so the number against is 4z. The survey states that 12,000 more people voted against, so the equation is 4z = z + 12,000. Solving for z gives 4,000.

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

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

Only \( (3, 3) \) is a solution that satisfies both equations.

The second equation is just a multiple of the first. Dividing the second equation by -2 yields \( 5x + 4y = 16 \). Thus, the two lines are the same and intersect at infinitely many points.

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

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

Testing each pair shows that only \( (4, 5) \) satisfies both equations.

The volume of a cube is \( V = s^3 \). Given \( V = 343 \), find \( s = \sqrt[3]{343} = 7 \). The surface area formula is \( 6s^2 \). Therefore, the surface area is \( 6(7)^2 = 294 \) square units.

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

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.

Solving for \( y \) using substitution or elimination, we find that \( y = 30 \).

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

In the function j(x) = -0.5x + 10, the y-intercept (where x = 0) represents the initial amount of water the runner had before starting the marathon. When x = 0, j(0) = 10, indicating that the runner started with 10 liters of water.