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

A scientist observes an initial population of 1,500 cells. Twelve hours later, the population grows to 24,000. Using the exponential growth formula \( P = C(2)^{rt} \), where \( P \) is the cell count at \( t \) hours, determine the value of \( r \).

The table below gives the coordinates of two points on a line in the xy-plane:

| x | y |
|----|----|
| \(p\) | -7 |
| \(p + 8\) | 21 |

The y-intercept of the line is at \((p - 6, b)\), where \(p\) and \(b\) are constants. What is the value of \(b\)?

Given \(f(x) = 3x^2 + 24x + 59\), the function \(g\) is defined by \(g(x) = f(x - 4)\). For what value of \(x\) does \(g(x)\) reach its minimum?

p(t) = 350 - 6t
The function p represents the volume of liquid, in ounces, in a glass t seconds after it starts spilling. According to the model, what is the predicted volume, in fluid ounces, spilling from the glass every half minute?

Poll results:
Candidate X - 350 votes
Candidate Y - 450 votes

In a poll of 800 voters, Candidate X received 350 votes, and Candidate Y received 450 votes. If 5,600 people vote in the election, how many more votes is Candidate Y expected to receive compared to Candidate X?

The growth of a bacterial culture is described by \( b(t) = 15,000 \cdot (1.06)^{t/100} \), with \( t \) in minutes. How much time, in hours, does it take for the bacteria count to double?

Triangles \( \triangle XYZ \) and \( \triangle ABC \) are congruent, where \( X \) corresponds to \( A \), and \( Y \) and \( B \) are right angles. If the measure of angle \( Z \) is 70°, what is the measure of angle \( C \)?

\(g(x) = 8000(0.75)^{x/6}\)

The function \(g\) gives the value, in dollars, of a certain piece of equipment after \(x\) months of use. If the value of the equipment decreases each year by \(q\)% of its value from the preceding year, what is the value of \(q\)?

A farm includes a 10-acre orchard and a 30-acre pasture. The total number of apple trees on the farm is 1,800. The equation 10𝑎 + 30𝑏 = 1,800 represents this situation. Which of the following is the best interpretation of 𝑎 in this context?

The population of a certain bacteria colony is initially 1,000. The population triples every hour. Which equation represents the population \( p \) after \( x \) days?

A colony of microorganisms starts with a population of 4,000. After three hours, the population has grown to 32,000. Following the exponential growth formula \( P = C(2)^{rt} \), where \( t \) represents hours, determine the value of \( r \).

\(x^2 - 12x + 11 = 0.\) One solution to the given equation can be written as \(6 + \sqrt{k}\), where \(k\) is a constant. What is the value of \(k\)?

Given the system of equations:

\( 20x + 3y = 150 \)
\( 4x + y = 30 \)

The solution to the system is \( (x, y) \). What is the value of \( y \)?

The equation describes the relationship between the number of fish, \( a \), and the number of turtles, \( c \), that a pet shop can care for. If the shop can care for 12 turtles on a given day, how many fish can it care for?

\( 1.5a + 4.5c = 90 \)

If \(\frac{c}{d} = 2\) and \(\frac{40c}{qd} = 2\), what is the value of \(q\)?

At how many points do the graphs of the given equations intersect in the xy-plane?

\( y = \frac{1}{2}x + 5 \)
\( y = -\frac{1}{2}x + 5 \)

Poll results:
Candidate A - 600 votes
Candidate B - 400 votes

In a random poll of 1,000 voters, the above results were recorded. If 10,000 people vote in the election, by how many votes is Candidate A expected to win?

g(t) = 500 - 7t
The function g models the volume of liquid, in milliliters, in a bottle t minutes after it starts leaking. According to the model, what is the predicted volume, in liters, leaking from the bottle each hour?

For \(f(x) = -3x^2 + 12x - 5\), let \(g(x) = f(x - 6)\). For what value of \(x\) does \(g(x)\) reach its maximum?

The following equation relates the variables \( x \) and \( y \):

\( y = x^2 - 8x + 18 \)

For what value of \( x \) does \( y \) reach its minimum?

A right triangle has legs with lengths of \( 9 , \text{cm} \) and \( 12 , \text{cm} \). If the length of the hypotenuse, in cm, can be written in the form \( 3\sqrt{d} \), where \( d \) is an integer, what is the value of \( d \)?

Given the system of equations:

\( 10x + 6y = 120 \)
\( 2x + y = 15 \)

The solution to the system is \( (x, y) \). What is the value of \( y \)?

Triangle ABC is similar to triangle DEF, where angle A corresponds to angle D and angles B and E are right angles. If \( \sin(A) = \frac{120}{125} \), what is the value of \( \sin(D) \)?

At how many points do the graphs of the given equations intersect in the xy-plane?

\( 5x + 4y = 16 \) and \( -10x - 8y = -32 \)

The half-life of a radioactive substance is 3 years. The initial quantity of the substance is 5 grams. Which equation best represents the quantity \( Q \) of the substance remaining after \( x \) minutes?

Triangles \( \triangle ABC \) and \( \triangle DEF \) are congruent, where \( A \) corresponds to \( D \), and \( B \) and \( E \) are right angles. If the measure of angle \( C \) is 55°, what is the measure of angle \( F \)?

\(x^2 - 8x + 7 = 0.\) One solution to the given equation can be written as \(4 + \sqrt{k}\), where \(k\) is a constant. What is the value of \(k\)?

If \(f(x) = x^2 - 10x + 25\), and \(g(x) = f(x - 2)\), for what value of \(x\) does \(g(x)\) reach its minimum?

The function \( f(x) = \frac{1}{16}(x - 5)^2 + 2 \) gives a roller coaster car's height above the ground \( f(x) \), in feet, \( x \) seconds after it started moving on a track, where \(\) 0 < x < 12 [/latex]. Which of the following is the best interpretation of the vertex of the graph of [latex] y = f(x) [/latex] in the [latex] xy [/latex]-plane?

For \(x > 0\), the function \(h\) is defined as follows: \(h(x)\) equals 80% of \(x\). Which of the following could describe this function?

The function \( f \) is defined by \( f(x) = 5x - 15 \). The graph of \( y = f(x) \) in the xy-plane has an x-intercept at \( (a, 0) \) and y-intercept at \( (0, b) \), where \( a \) and \( b \) are constants. What is the value of \( a + b \)?

At how many points do the graphs of the given equations intersect in the xy-plane?

\( 3x + 5y = 25 \) and \( 6x + 10y = 50 \)

Triangles \( \triangle LMN \) and \( \triangle PQR \) are congruent, where \( L \) corresponds to \( P \), and \( M \) and \( Q \) are right angles. The measure of angle \( L \) is 30°. What is the measure of angle \( R \)?

The measure of angle S is \( \frac{2\pi}{3} \) radians. The measure of angle T is \( \frac{\pi}{4} \) radians greater than the measure of angle S. What is the measure of angle T, in degrees?

A cube has a volume of 729 cubic units. What is the surface area, in square units, of the cube?

A garden contains a 6-square-meter vegetable patch and a 12-square-meter flower bed. The total number of plants in the garden is 216. The equation 6𝑣 + 12𝑓 = 216 represents this situation. Which of the following is the best interpretation of 𝑣 in this context?

The function \( f(x) = \frac{1}{9}(x - 7)^2 + 3 \) gives a metal ball's height above the ground \( f(x) \), in inches, \( x \) seconds after it started moving on a track, where \(\) 0 < x < 10 [/latex]. Which of the following is the best interpretation of the vertex of the graph of [latex] y = f(x) [/latex] in the [latex] xy [/latex]-plane?

If \( 39 \) is \( p \% \) of \( 65 \), what is \( p \% \) of \( 39 \)?

Caleb used juice to make popsicles. The function f(x) = -5x + 30 approximates the volume, in fluid ounces, of juice Caleb had remaining after making x popsicles. Which statement is the best interpretation of the y-intercept of the graph of y=f(x) in the xy-plane in this context?

At how many points do the graphs of the given equations intersect in the xy-plane?

\( y = 4x + 8 \)
\( y = 4x - 2 \)

In the given equation, \( (7x + p)(5x^2 - 25)(4x^2 - 14x + 5p) = 0 \), where \( p \) is a positive constant. The sum of the solutions to the equation is \( 10 \). What is the value of \( p \)?

The function \( f \) is defined by \( f(x) = 180(0.3)^x \). What is the value of \( f(0) \)?

In the given equation, \( (3x + p)(5x^2 - 45)(3x^2 - 16x + 6p) = 0 \), where \( p \) is a positive constant. The sum of the solutions to the equation is \( \frac{20}{3} \). What is the value of \( p \)?

Triangles \( \triangle PQR \) and \( \triangle STU \) are congruent, where \( P \) corresponds to \( S \), and \( Q \) and \( T \) are right angles. The measure of angle \( P \) is 25°. What is the measure of angle \( U \)?