All practice questions

Question 1981Question →

A hybrid car's fuel tank has a capacity of $12$ gallons. When driving on a highway, the amount of fuel in the tank decreases at a constant rate. After driving for $1.5$ hours, $9.6$ gallons of fuel remain in the tank. If the amount of fuel in the tank, $F$ , in gallons, after driving for $t$ hours is modeled by a linear equation, what is the value of $F$ when $t = 4$ ?

Show answer & explanation

Answer: 5.6

Answer

The amount of fuel remaining in the tank after driving for

4

hours is

5.6

gallons.

The amount of fuel in the tank,

F

, and the driving time,

t

, share a linear relationship. The initial amount of fuel at

t = 0

is

12

gallons, representing the vertical intercept. The fuel decreases at a constant rate, which is the slope of the line. Over a period of

1.5

hours, the amount of fuel decreases by

12 - 9.6 = 2.4

gallons. The rate of decrease is

\frac{2.4}{1.5} = 1.6

gallons per hour, so the slope is

-1.6

. The linear model is

F = -1.6t + 12

. Substituting

t = 4

into this equation yields

F = -1.6(4) + 12 = -6.4 + 12 = 5.6

gallons.

Step-by-Step Solution

1

Calculate the constant rate of fuel consumption (the slope of the linear relationship).

The rate of consumption is

1.6

gallons per hour.

Since the fuel decreases at a constant rate, the change in fuel divided by the change in time gives the rate of consumption. In

1.5

hours, the fuel decreases from

12

gallons to

9.6

gallons, which is a decrease of

12 - 9.6 = 2.4

gallons. Thus, the rate of consumption is

\frac{2.4\text{ gallons}}{1.5\text{ hours}} = 1.6

gallons per hour.

2

Write the linear equation modeling the fuel remaining in the tank,

F

, as a function of time,

t

.

F = -1.6t + 12

The initial amount of fuel when

t = 0

is

12

gallons, which represents the vertical intercept (

b = 12

). The fuel decreases at a constant rate of

1.6

gallons per hour, which represents a slope of

m = -1.6

.

3

Substitute

t = 4

into the linear equation to find the value of

F

.

F = 5.6

Evaluating the equation at

t = 4

yields

F = -1.6(4) + 12 = -6.4 + 12 = 5.6

.

Key Concept

Linear Equations in Two Variables

Question 1982Question →

The table below shows some values for a cubic polynomial function $g$ .

$x$	$g(x)$
$-1$	$0$
$1$	$16$
$3$	$0$

In the $xy$ -plane, the graph of $y = g(x)$ is tangent to the $x$ -axis at $x = 3$ . What is the value of $g(0)$ ?

Show answer & explanation

Answer: 18

Answer

18

The correct value of

g(0)

is 18. By using the fact that

g(-1) = 0

, we establish

(x + 1)

as a factor. The tangency at

x = 3

tells us that

(x - 3)^2

is a factor. Writing the function as

g(x) = a(x + 1)(x - 3)^2

and substituting

g(1) = 16

gives

8a = 16 \implies a = 2

. Evaluating

g(0)

yields

2(1)(9) = 18

.

Step-by-Step Solution

1

Determine the factors of the cubic polynomial

g(x)

using the given zeros and the tangency condition.

The factors are

(x + 1)

and

(x - 3)^2

, so the function is of the form

g(x) = a(x + 1)(x - 3)^2

.

Since

g(-1) = 0

,

x = -1

is a root of the polynomial. The graph being tangent to the

x

-axis at

x = 3

indicates that

x = 3

is a root with a multiplicity of at least 2. Since

g(x)

is a cubic polynomial (degree 3), the multiplicity of the root at

x = 3

must be exactly 2.

2

Use the table value

g(1) = 16

to solve for the constant coefficient

a

.

a = 2

Substituting

x = 1

into

g(x) = a(x + 1)(x - 3)^2

gives

g(1) = a(1 + 1)(1 - 3)^2 = 8a

. Setting this equal to the table value of 16 yields

8a = 16

, which simplifies to

a = 2

.

3

Evaluate the polynomial at

x = 0

using the fully determined function

g(x) = 2(x + 1)(x - 3)^2

.

18

To find

g(0)

, substitute

x = 0

into the expression:

g(0) = 2(0 + 1)(0 - 3)^2 = 2(1)(9) = 18

.

Key Concept

Identifying polynomial factors from graphs and tables, and analyzing root multiplicity (tangency vs. crossing).

Question 1983Question →

A right circular cylinder has a volume of $20\text{ cubic inches}$ . If the radius of the cylinder is tripled and its height remains the same, what is the volume, in cubic inches, of the new cylinder?

60

120

180

540

Show answer & explanation

Answer: 180

Answer

180

The volume of a right circular cylinder is calculated using the formula

V = \pi r^2 h

. When the radius

r

is tripled to

3r

and the height

h

remains constant, the volume is multiplied by

3^2 = 9

. Thus, the new volume is

9

times the original volume of

20

cubic inches, which equals

180

cubic inches.

Step-by-Step Solution

1

Write down the formula for the volume of a right circular cylinder.

V = \pi r^2 h

To identify how the volume of a cylinder relates to its radius and height.

2

Determine the effect of tripling the radius on the volume formula.

V_{\text{new}} = \pi (3r)^2 h = 9\pi r^2 h

Substituting the new radius

3r

into the formula shows that the new volume is 9 times the original volume.

3

Substitute the original volume of

20

cubic inches into the expression for the new volume.

V_{\text{new}} = 9 \times 20 = 180

Since the original volume

\pi r^2 h

equals

20

, we multiply this value by 9 to find the new volume.

Key Concept

Volume of a cylinder and dimensional scaling

Estimated Time:45s

Question 1984Question →

The quadratic equation $x^2 - 12x + 4 = 0$ has solutions $x_1$ and $x_2$ . What is the value of $\frac{1}{x_1} + \frac{1}{x_2}$ ?

Show answer & explanation

Answer: 3

Answer

The correct answer is 3.

By writing the expression

\frac{1}{x_1} + \frac{1}{x_2}

with a common denominator, we get

\frac{x_1 + x_2}{x_1 x_2}

. For the quadratic equation

x^2 - 12x + 4 = 0

, Vieta's formulas give the sum of the roots

x_1 + x_2 = 12

and the product of the roots

x_1 x_2 = 4

. Substituting these values into the fraction yields

\frac{12}{4} = 3

.

Step-by-Step Solution

1

Find a common denominator to combine the terms in the given expression.

\frac{1}{x_1} + \frac{1}{x_2} = \frac{x_1 + x_2}{x_1 x_2}

To express the target quantity in terms of the sum and product of the quadratic solutions.

2

Apply Vieta's formulas to find the sum and product of the solutions from the quadratic equation

x^2 - 12x + 4 = 0

.

x_1 + x_2 = 12

and

x_1 x_2 = 4

For any quadratic equation

ax^2 + bx + c = 0

, the sum of the roots is

-\frac{b}{a}

and the product is

\frac{c}{a}

.

3

Substitute the sum and product values into the combined fraction.

\frac{12}{4} = 3

To compute the numerical value of the expression.

Key Concept

Sum and product of solutions of a quadratic equation

Question 1985Question →

The graph of the quadratic function $f$ in the $xy$ -plane has $x$ -intercepts at $(-2, 0)$ and $(8, 0)$ . If the maximum value of $f(x)$ is $25$ , what is the value of $f(0)$ ?

Show answer & explanation

Answer: 16

Answer

16

The axis of symmetry of the quadratic function

f

lies halfway between the

x

-intercepts

x = -2

and

x = 8

, which is at

x = \frac{-2 + 8}{2} = 3

. Since the function has a maximum value of

25

, this maximum must occur at the vertex, giving the vertex coordinates

(3, 25)

. In vertex form, the function is

f(x) = a(x - 3)^2 + 25

. Substituting the

x

-intercept

(8, 0)

into the function yields

0 = a(8 - 3)^2 + 25

, which simplifies to

25a = -25

, or

a = -1

. Therefore, the equation of the function is

f(x) = -(x - 3)^2 + 25

. Evaluating this at

x = 0

gives

f(0) = -(0 - 3)^2 + 25 = -9 + 25 = 16

.

Step-by-Step Solution

1

Find the

x

-coordinate of the vertex (axis of symmetry)

x = 3

The axis of symmetry of a parabola is located exactly halfway between its

x

-intercepts:

x = \frac{-2 + 8}{2} = 3

.

2

Determine the vertex coordinates

(3, 25)

The maximum value of the quadratic function occurs at its vertex, so the

y

-coordinate of the vertex is the maximum value

25

.

3

Write the vertex form of the quadratic function

f(x) = a(x - 3)^2 + 25

The vertex form of a quadratic function is

f(x) = a(x - h)^2 + k

, where

(h, k)

is the vertex.

4

Solve for the leading coefficient

a

a = -1

Substitute the

x

-intercept

(8, 0)

into the vertex form:

0 = a(8 - 3)^2 + 25 \implies 25a = -25 \implies a = -1

.

5

Find the value of

f(0)

f(0) = 16

Substitute

x = 0

into the function:

f(0) = -(0 - 3)^2 + 25 = -9 + 25 = 16

.

Key Concept

Using

x

-intercepts and the maximum value to determine the vertex and equation of a quadratic function.

Question 1986Question →

If $x$ satisfies the equation below, what is the value of $x + 5$ ?

x - 4 = \sqrt{4x + 5}

Show answer & explanation

Answer: 16

Answer

The correct answer is 16.

Squaring both sides of the equation

x - 4 = \sqrt{4x + 5}

results in

(x - 4)^2 = 4x + 5

, which expands to

x^2 - 8x + 16 = 4x + 5

. Subtracting

4x + 5

from both sides gives the quadratic equation

x^2 - 12x + 11 = 0

. Factoring this quadratic equation yields

(x - 11)(x - 1) = 0

, giving candidate solutions of

x = 11

and

x = 1

. Checking these solutions in the original equation shows that

x = 11

is valid (

7 = 7

), whereas

x = 1

is extraneous (

-3 = 3

is false). Therefore, the only real solution is

x = 11

, and the value of the expression

x + 5

is

11 + 5 = 16

.

Step-by-Step Solution

1

Square both sides of the equation to remove the radical.

(x - 4)^2 = 4x + 5

Squaring a square root isolates the expression under the radical.

2

Expand the squared binomial on the left side.

x^2 - 8x + 16 = 4x + 5

Applying the distributive property to

(x - 4)(x - 4)

yields

x^2 - 8x + 16

.

3

Rearrange the equation to set it equal to zero.

x^2 - 12x + 11 = 0

Subtracting

4x + 5

from both sides simplifies the equation into standard quadratic form.

4

Factor the quadratic equation.

(x - 11)(x - 1) = 0

Finding two numbers that multiply to 11 and add to -12 gives -11 and -1.

5

Solve for the candidate values of x.

x = 11

and

x = 1

Setting each factor equal to zero yields the possible solutions.

6

Check the candidate values in the original equation to identify any extraneous solutions.

x = 11

is valid;

x = 1

is extraneous.

Substituting

x = 1

results in

-3 = 3

, which is false because the principal square root is always non-negative.

7

Evaluate the target expression using the valid solution.

11 + 5 = 16

The question asks for the value of

x + 5

, so we substitute the only valid solution,

x = 11

.

Key Concept

Solving radical equations by squaring both sides and checking for extraneous solutions.

Question 1987Question →

A municipal parking garage tracked the charging times of $200$ electric vehicles over a one-week period. The vehicles were classified by type—Battery Electric Vehicles (BEVs) or Plug-in Hybrid Electric Vehicles (PHEVs)—and by the time of day they primarily charged: Daytime (6:00 AM to 6:00 PM) or Nighttime (6:00 PM to 6:00 AM). The results are summarized in the table below.

Vehicle Type	Daytime Charging	Nighttime Charging	Total
BEV	$45$	$75$	$120$
PHEV	$55$	$25$	$80$
Total	$100$	$100$	$200$

If one of these electric vehicles is selected at random, and it is found to have charged during the nighttime, what is the probability that the vehicle is a PHEV?

\frac{1}{8}

\frac{1}{4}

\frac{5}{16}

\frac{3}{8}

Show answer & explanation

Answer: $\frac{1}{4}$

Answer

\frac{1}{4}

The correct answer is the simplified fraction representing the number of nighttime-charging PHEVs (

25

) divided by the total number of nighttime-charging vehicles (

100

), which is

\frac{25}{100} = \frac{1}{4}

.

Step-by-Step Solution

1

Identify the total number of vehicles in the restricted sample space.

The total number of vehicles that charged during the nighttime is

100

.

The question specifies that the vehicle 'is found to have charged during the nighttime', which limits the denominator to the 'Nighttime Charging' column total.

2

Determine the number of favorable outcomes within this restricted sample space.

The number of PHEVs that charged during the nighttime is

25

.

We need to find the count in the intersection of the 'PHEV' row and the 'Nighttime Charging' column.

3

Calculate the conditional probability as a fraction.

The probability is

\frac{25}{100} = \frac{1}{4}

.

Dividing the number of favorable vehicles (

25

) by the conditional total (

100

) yields the final probability, which simplifies to

\frac{1}{4}

.

Key Concept

Conditional probability calculations from a two-way table involve restricting the sample space to a specific row or column total rather than the grand total.

Estimated Time:1m 30s

Question 1988Question →

A right circular cone has a volume of $24\text{ cubic centimeters}$ . A second right circular cone has a base radius that is $3$ times the base radius of the first cone, and a height that is half the height of the first cone. What is the volume, in cubic centimeters, of the second cone?

36

108

216

324

Show answer & explanation

Answer: 108

Answer

108

The volume of a cone is proportional to the square of its radius and directly proportional to its height. Tripling the radius increases the volume by a factor of

3^2 = 9

. Halving the height reduces the volume by a factor of

\frac{1}{2}

. Combining these changes scales the original volume by

9 \times \frac{1}{2} = 4.5

. Multiplying the original volume of

24

by

4.5

gives the correct volume of

108

.

Step-by-Step Solution

1

Write down the formula for the volume of a right circular cone.

V = \frac{1}{3}\pi r^2 h

This establishes the relationship between volume, radius, and height.

2

Express the volume of the second cone (

V_2

) in terms of the dimensions of the first cone (

r

and

h

).

V_2 = \frac{1}{3}\pi (3r)^2 \left(\frac{1}{2}h\right)

The radius of the second cone is

3r

and the height is

\frac{1}{2}h

.

3

Simplify the expression for the second cone's volume to find the scale factor relative to the first cone.

V_2 = \frac{1}{3}\pi (9r^2) \left(\frac{1}{2}h\right) = \frac{9}{2} \left(\frac{1}{3}\pi r^2 h\right) = 4.5 V_1

Squaring the radius multiplier of 3 gives a factor of 9, and the height multiplier of 0.5 gives a factor of 0.5, resulting in a combined scaling factor of 4.5.

4

Calculate the final volume using the volume of the first cone (

V_1 = 24

).

V_2 = 4.5 \times 24 = 108

Multiplying the original volume of

24\text{ cm}^3

by the overall scale factor of

4.5

yields the new volume.

Key Concept

Dimensional scaling of three-dimensional solids

Question 1989Question →

If $9^{2x + 1} = \left(\frac{1}{27}\right)^{x - 2}$ , what is the value of $x$ ?

-8

1

\frac{4}{7}

-\frac{8}{7}

Show answer & explanation

Answer: $\frac{4}{7}$

Answer

\frac{4}{7}

The correct answer is

\frac{4}{7}

. By rewriting both sides of the equation with a common base of

3

, the equation becomes

3^{2(2x + 1)} = 3^{-3(x - 2)}

. Applying the power of a power rule gives

3^{4x + 2} = 3^{-3x + 6}

. Since the bases are equal, the exponents must be equal, giving

4x + 2 = -3x + 6

. Solving for

x

results in

7x = 4

, which gives

x = \frac{4}{7}

.

Step-by-Step Solution

1

Rewrite each side of the equation with a common base of

3

.

9 = 3^2

and

\frac{1}{27} = 3^{-3}

, so the equation becomes

(3^2)^{2x + 1} = (3^{-3})^{x - 2}

.

Before solving an exponential equation, it is helpful to express the bases in terms of their common prime base.

2

Apply the exponent power rule

(a^m)^n = a^{mn}

to simplify the exponents.

3^{2(2x + 1)} = 3^{-3(x - 2)} \implies 3^{4x + 2} = 3^{-3x + 6}

.

Simplifying the expressions on both sides allows for equating the exponents directly.

3

Set the exponents equal to each other and solve the resulting linear equation.

4x + 2 = -3x + 6 \implies 7x = 4 \implies x = \frac{4}{7}

.

Since the bases are equal, the powers can only be equal if their exponents are equal.

Key Concept

Solving exponential equations by expressing bases in terms of a common base and equating the exponents.

Estimated Time:1m 30s

Question 1990Question →

A city reservoir contains water that is released at a constant rate for river conservation. The volume of water, $W$ , in millions of gallons, remaining in the reservoir $d$ days after the release begins is modeled by the equation $W = 240 - 1.8d$ . What is the best interpretation of the number $1.8$ in this context?

The reservoir loses 1 million gallons of water every 1.8 days.

The initial volume of water in the reservoir before the release begins is 1.8 million gallons.

The volume of water in the reservoir decreases by 1.8 million gallons each day.

The total duration of the controlled release is 1.8 days.

Show answer & explanation

Answer: The volume of water in the reservoir decreases by 1.8 million gallons each day.

Answer

The volume of water in the reservoir decreases by 1.8 million gallons each day.

The equation

W = 240 - 1.8d

is in slope-intercept form, where the slope is

-1.8

and the

y

-intercept is

240

. The slope represents the rate of change of the remaining volume of water with respect to time. A slope of

-1.8

indicates that the volume of water,

W

, decreases by

1.8

million gallons for every

1

day increase in the number of days,

d

.

Step-by-Step Solution

1

Identify the variables and constants in the given linear model.

In

W = 240 - 1.8d

,

W

is the remaining volume in millions of gallons,

d

is the time in days, 240 is the y-intercept (initial volume), and -1.8 is the slope (rate of change).

Understanding the components of a linear equation

y = mx + b

helps isolate the meaning of each constant.

2

Interpret the meaning of the slope in the context of the variables and units.

The coefficient of

d

is -1.8, which represents a rate of change of -1.8 million gallons per day.

The slope is the change in the dependent variable (

W

, in millions of gallons) per unit change in the independent variable (

d

, in days).

3

Relate the negative sign to the physical context.

A negative slope of -1.8 means that the volume decreases by 1.8 million gallons for each elapsed day.

In context, a constant release of water leads to a decrease in the remaining volume.

Key Concept

Interpreting slope in a linear relationship context

Estimated Time:1m 30s

Question 1991Question →

In a laboratory experiment, a wet soil sample has a mass of $800$ grams. During a drying process, the sample loses $20\%$ of its mass. After the drying process, a nutrient solution is added, which increases the sample's mass by $15\%$ of its dried mass. What is the final mass, in grams, of the soil sample after the solution is added?

544

655

736

760

Show answer & explanation

Answer: 736

Answer

736

To find the final mass of the soil sample, first calculate the mass after it loses

20\%

of its initial

800

grams:

800 \times (1 - 0.20) = 640

grams. Next, calculate the final mass after a

15\%

increase is applied to this intermediate mass:

640 \times (1 + 0.15) = 736

grams. This shows that the final mass of the sample is

736

grams.

Step-by-Step Solution

1

Calculate the mass of the soil sample after losing

20\%

of its initial mass of

800

grams.

Dried mass =

800 \times (1 - 0.20) = 640

grams.

This establishes the intermediate baseline mass before the nutrient solution is added.

2

Calculate the final mass of the soil sample after a

15\%

increase is applied to the dried mass of

640

grams.

Final mass =

640 \times (1 + 0.15) = 736

grams.

The

15\%

increase must be calculated using the dried mass of

640

grams as the new base.

Key Concept

Multi-step percent change and identifying the correct base value for sequential percentage increases or decreases.

Alternative Method

You can express the sequential percent changes as a single product:

800 \times 0.80 \times 1.15

. Multiplying these values directly yields

800 \times 0.92 = 736

grams.

Estimated Time:1m 15s

Question 1992Question →

A botanist conducted an experiment to study the germination rates of seeds in two different soil mixtures, Mixture A and Mixture B. The results of the experiment are partially shown in the table below, where $x$ is a positive constant.

Soil Mixture	Germinated	Did Not Germinate
Mixture A	$x$	$12$
Mixture B	$30$	$x + 6$

Of the seeds in the experiment that did not germinate, the probability that the seed was planted in Mixture A is $\frac{1}{3}$ . Given that a seed selected at random from the experiment germinated, what is the probability that it was planted in Mixture A?

Show answer & explanation

Answer: 0.375

Answer

3/8 (or 0.375)

To find the probability that a germinated seed was planted in Mixture A, we must first solve for the variable

x

. The number of seeds in Mixture A that did not germinate is

12

, and the total number of seeds that did not germinate is

12 + (x + 6) = x + 18

. We are given that the probability a non-germinated seed was planted in Mixture A is

\frac{1}{3}

. Setting up the equation:

\frac{12}{x + 18} = \frac{1}{3}

yields

x + 18 = 36

, so

x = 18

. Substituting this value back into the table, the number of germinated seeds in Mixture A is

18

. The total number of germinated seeds is the sum of germinated seeds in Mixture A and Mixture B:

18 + 30 = 48

. Thus, the probability that a germinated seed was planted in Mixture A is

\frac{18}{48}

, which simplifies to

\frac{3}{8}

(or

0.375

).

Step-by-Step Solution

1

Set up the conditional probability equation for seeds that did not germinate to find

x

.

x = 18

From the table, the number of seeds in Mixture A that did not germinate is

12

, and the number of seeds in Mixture B that did not germinate is

x + 6

. The total number of seeds that did not germinate is

12 + (x + 6) = x + 18

. The probability that a seed that did not germinate was from Mixture A is

\frac{12}{x + 18} = \frac{1}{3}

. Solving this equation gives

36 = x + 18

, which simplifies to

x = 18

.

2

Find the number of germinated seeds in Mixture A and the total number of germinated seeds.

Mixture A germinated =

18

, Total germinated =

48

Substituting

x = 18

, the number of germinated seeds in Mixture A is

18

. The number of germinated seeds in Mixture B is given as

30

. The total number of germinated seeds is

18 + 30 = 48

.

3

Calculate the conditional probability that a germinated seed was planted in Mixture A.

\frac{3}{8}

or

0.375

The probability is the number of germinated seeds in Mixture A divided by the total number of germinated seeds, which is

\frac{18}{48} = \frac{3}{8}

, or

0.375

as a decimal.

Key Concept

Conditional Probability in Two-Way Tables

Question 1993Question →

A right circular cylinder has a height of $12$ and a base radius of $r$ . A sphere has a radius of $r$ . If the volume of the cylinder is equal to the volume of the sphere, what is the value of $r$ ?

Show answer & explanation

Answer: 9

Answer

9

The volume of a cylinder is given by

V = \pi r^2 h

and the volume of a sphere is given by

V = \frac{4}{3}\pi r^3

. Given that the cylinder's height is

12

, its volume is

12\pi r^2

. Setting the volumes equal yields

12\pi r^2 = \frac{4}{3}\pi r^3

. Since

r

is a non-zero radius, we can divide both sides by

\pi r^2

, resulting in

12 = \frac{4}{3}r

. Multiplying both sides by

\frac{3}{4}

gives

r = 9

.

Step-by-Step Solution

1

State the standard volume formulas for a right circular cylinder and a sphere.

V_{\text{cylinder}} = \pi r^2 h

and

V_{\text{sphere}} = \frac{4}{3}\pi r^3

These formulas are needed to express the volumes of both solids in terms of

r

.

2

Equate the volume of the cylinder to the volume of the sphere and substitute the given height of

12

.

12\pi r^2 = \frac{4}{3}\pi r^3

The problem states that the volume of the cylinder is equal to the volume of the sphere.

3

Divide both sides of the equation by

\pi r^2

.

12 = \frac{4}{3}r

Simplifies the equation to a first-degree equation in terms of

r

.

4

Solve for

r

by multiplying both sides of the simplified equation by the reciprocal of

\frac{4}{3}

, which is

\frac{3}{4}

.

r = 9

Isolates the variable

r

to find the correct value.

Key Concept

Equating the volumes of geometric solids to solve for an unknown dimension.

Estimated Time:1m 30s

Question 1994Question →

An object is launched from a platform. The function $h(t) = -5t^2 + 30t + 12$ models the height of the object, in meters, $t$ seconds after it was launched. What is the maximum height, in meters, reached by the object?

3

12

57

147

Show answer & explanation

Answer: 57

Answer

The maximum height reached by the object is 57 meters.

The given function

h(t) = -5t^2 + 30t + 12

represents a parabola opening downward because the coefficient of

t^2

is negative. The maximum value of this function occurs at its vertex. The time

t

at the vertex is determined using the formula

t = -\frac{b}{2a} = -\frac{30}{2(-5)} = 3

seconds. Substituting

t = 3

back into the function yields the maximum height:

h(3) = -5(3)^2 + 30(3) + 12 = -45 + 90 + 12 = 57

meters.

Step-by-Step Solution

1

Identify the coefficients of the quadratic function in standard form

h(t) = at^2 + bt + c

.

a = -5

,

b = 30

, and

c = 12

.

These coefficients are needed to calculate the vertex of the parabola.

2

Calculate the time

t

at which the maximum height occurs using the vertex formula

t = -\frac{b}{2a}

.

t = -\frac{30}{2(-5)} = 3

seconds.

Since the leading coefficient

a = -5

is negative, the parabola opens downward, meaning its vertex represents the maximum value.

3

Substitute

t = 3

back into the height function to find the maximum height.

h(3) = -5(3)^2 + 30(3) + 12 = -45 + 90 + 12 = 57

meters.

Evaluating the function at the time of the vertex gives the corresponding maximum height.

Key Concept

The maximum value of a quadratic function

f(x) = ax^2 + bx + c

(where

a < 0

) is the vertical coordinate of its vertex, which occurs at

x = -\frac{b}{2a}

.

Question 1995Question →

A researcher at a university wants to estimate the average number of hours that students at the university spend studying per week. The researcher randomly selects $200$ students from a database of all students enrolled at the university to participate in a survey. Which of the following is the largest population to which the results of the survey can be most reasonably generalized?

All students enrolled at the university

All college students in the nation

Only the 200 students who were selected for the survey

All students who study more than the average number of hours

Show answer & explanation

Answer: All students enrolled at the university

Answer

All students enrolled at the university

The correct option is the one stating that the results can be generalized to all students enrolled at the university. In an observational study or survey, results can be generalized to the population from which the sample was randomly selected. Here, the database used for selection included all students enrolled at the university, making them the target population.

Step-by-Step Solution

1

Identify the population from which the sample was drawn.

The sample of

200

students was randomly selected from a database containing all students enrolled at the university.

Generalization of survey results requires that the sample be randomly selected from the target population.

2

Determine the scope of the generalization based on the random sampling design.

The results can be generalized to the entire population of students enrolled at the university, but not to broader populations (like all college students in the nation) or restricted biased subgroups.

Random selection ensures that the sample is representative of the population from which it was drawn.

Key Concept

Generalizing results from a random sample to the parent population

Question 1996Question →

The graph of the function $f$ in the $xy$ -plane has a vertex at $(-2, 7)$ . The function $g$ is defined by $g(x) = f(-x + 3) + 12$ . If the vertex of the graph of $y = g(x)$ is the point $(a, b)$ , what is the value of $a + b$ ?

Show answer & explanation

Answer: 24

Answer

24

The vertex of the parent function

f(x)

is at

(-2, 7)

, which means

f(-2) = 7

. The transformed function is

g(x) = f(-x + 3) + 12

. The vertex of

g(x)

occurs when the input to

f

, which is

-x + 3

, is equal to

-2

. Solving

-x + 3 = -2

gives

-x = -5

, or

x = 5

, so the x-coordinate of the vertex of

g(x)

is

a = 5

. To find the y-coordinate

b

, we evaluate

g(5) = f(-(5) + 3) + 12 = f(-2) + 12 = 7 + 12 = 19

. Therefore, the vertex of the graph of

y = g(x)

is

(5, 19)

, so

a = 5

and

b = 19

. The value of

a + b

is

5 + 19 = 24

.

Step-by-Step Solution

1

Find the x-coordinate of the vertex of the transformed function

g(x)

by setting the inner expression equal to the x-coordinate of the original vertex.

a = 5

The vertex of

f(x)

is located at

x = -2

. For

g(x) = f(-x + 3) + 12

, the vertex occurs when the input to

f

,

-x + 3

, is equal to

-2

. Solving

-x + 3 = -2

yields

x = 5

.

2

Find the y-coordinate of the vertex of

g(x)

by evaluating

g(5)

.

b = 19

Substituting

x = 5

into the definition of

g(x)

gives

g(5) = f(-2) + 12

. Since the vertex of

f

is at

(-2, 7)

,

f(-2) = 7

. Thus,

g(5) = 7 + 12 = 19

.

3

Calculate the sum of the coordinates

a

and

b

.

24

The vertex of

g(x)

is

(5, 19)

, so

a = 5

and

b = 19

. The sum

a + b

is

5 + 19 = 24

.

Key Concept

Determining the coordinates of a transformed vertex using function notation.

Question 1997Question →

A shipping container initially holds 450 packages. A crew unloads the container at a rate of 15 packages per hour, while an automated sorting machine unloads packages at a rate of $x$ packages per hour. The number of packages remaining in the container after 8 hours must be at most 130. The inequality $450 - 8(15 + x) \leq 130$ models this scenario. What is the minimum possible value of $x$ ?

Show answer & explanation

Answer: 25

Answer

The minimum possible value of

x

is 25.

To find the minimum possible value of

x

, solve the inequality

450 - 8(15 + x) \leq 130

. First, distribute the

-8

to obtain

450 - 120 - 8x \leq 130

. Simplify the constant terms on the left side to get

330 - 8x \leq 130

. Subtract 330 from both sides, yielding

-8x \leq -200

. Finally, divide both sides by

-8

and reverse the inequality sign because of the division by a negative number, which results in

x \geq 25

. The minimum possible value of

x

is 25.

Step-by-Step Solution

1

Distribute the factor of

-8

to both terms inside the parentheses.

450 - 120 - 8x \leq 130

To remove the parentheses and prepare to combine like terms.

2

Subtract 120 from 450 to simplify the constants on the left side.

330 - 8x \leq 130

To simplify the left side of the inequality before isolating the variable.

3

Subtract 330 from both sides of the inequality.

-8x \leq -200

To isolate the variable term on the left-hand side.

4

Divide both sides by

-8

and reverse the direction of the inequality symbol.

x \geq 25

Dividing by a negative value reverses the inequality sign. Since

x

must be greater than or equal to 25, the minimum possible value is 25.

Key Concept

Solving multi-step linear inequalities in one variable, including distributing coefficients and reversing the inequality sign when multiplying or dividing by a negative number.

Question 1998Question →

A sphere has a surface area of $144\pi$ square inches. What is the volume, in cubic inches, of the sphere?

48\pi

216\pi

288\pi

864\pi

Show answer & explanation

Answer: $288\pi$

Answer

The volume of the sphere is

288\pi

cubic inches.

The surface area of a sphere is given by

A = 4\pi r^2

, where

r

is the radius. Setting this equal to the given surface area yields

4\pi r^2 = 144\pi

, which simplifies to

r^2 = 36

, meaning the radius of the sphere is

6

inches. The volume of a sphere is given by

V = \frac{4}{3}\pi r^3

. Substituting

r = 6

into this formula gives

V = \frac{4}{3}\pi (6^3) = \frac{4}{3}\pi (216) = 288\pi

cubic inches.

Step-by-Step Solution

1

Set the formula for the surface area of a sphere,

A = 4\pi r^2

, equal to the given surface area of

144\pi

.

4\pi r^2 = 144\pi

, which simplifies to

r^2 = 36

.

To find the square of the radius of the sphere.

2

Take the square root of both sides to find the radius

r

.

r = 6

.

The radius of a sphere must be a positive number.

3

Substitute the radius

r = 6

into the formula for the volume of a sphere,

V = \frac{4}{3}\pi r^3

.

V = \frac{4}{3}\pi(6^3) = \frac{4}{3}\pi(216) = 288\pi

.

To compute the final volume of the sphere.

Key Concept

The volume of a sphere can be calculated by first finding its radius from its surface area using the formulas

A = 4\pi r^2

and

V = \frac{4}{3}\pi r^3

.

Question 1999Question →

In the $xy$ -plane, the graph of the quadratic function $f(x) = -x^2 + bx + c$ , where $b$ and $c$ are constants, has its vertex at $(4, 25)$ . If the positive $x$ -intercept of the graph of $f$ is $(d, 0)$ , what is the value of $d$ ?

Show answer & explanation

Answer: 9

Answer

The value of

d

is

9

.

The vertex form of a quadratic function is

f(x) = a(x-h)^2 + k

, where

(h, k)

is the vertex. Since the vertex is

(4, 25)

and the coefficient of

x^2

is

-1

, the function is

f(x) = -(x-4)^2 + 25

. Setting

f(x) = 0

to find the

x

-intercepts yields

-(x-4)^2 + 25 = 0

, which simplifies to

(x-4)^2 = 25

. Taking the square root of both sides gives

x - 4 = 5

or

x - 4 = -5

. Solving these equations gives

x = 9

or

x = -1

. The positive

x

-intercept is

(9, 0)

, so the value of

d

is

9

.

Step-by-Step Solution

1

Write the function in vertex form.

f(x) = -(x-4)^2 + 25

The vertex form of a quadratic function is

f(x) = a(x-h)^2 + k

, where

(h, k)

is the vertex. Here, the vertex is

(4, 25)

and the coefficient of

x^2

is

-1

, so

a = -1

,

h = 4

, and

k = 25

.

2

Set

f(x) = 0

to find the

x

-intercepts.

-(x-4)^2 + 25 = 0

The

x

-intercepts of a graph are the points where the function value is equal to 0.

3

Solve the equation for

x

.

x = 9

or

x = -1

Rearranging the equation gives

(x-4)^2 = 25

. Taking the square root of both sides gives

x-4 = 5

or

x-4 = -5

, which solves to

x = 9

or

x = -1

.

4

Identify the positive

x

-intercept coordinate

d

.

d = 9

The question asks for the positive

x

-intercept

(d, 0)

, which corresponds to

x = 9

.

Key Concept

Vertex form of a quadratic function and finding

x

-intercepts

Question 2000Question →

A dataset consists of 10 positive integers: $4, 6, 8, 8, 10, 12, 12, 14, 16$ , and $x$ . If the median of the dataset is equal to the mean of the dataset, and $x > 16$ , what is the value of $x$ ?

Show answer & explanation

Answer: 20

Answer

20

The correct answer is 20. When the 10 positive integers are sorted in ascending order, the condition

x > 16

ensures that

x

is the largest value and occupies the final position:

4, 6, 8, 8, 10, 12, 12, 14, 16, x

. The median of this 10-value dataset is the average of the 5th and 6th values, which are 10 and 12. Thus, the median is

\frac{10 + 12}{2} = 11

. The mean of the dataset is the sum of the 10 values divided by 10. The sum of the 9 known values is

4 + 6 + 8 + 8 + 10 + 12 + 12 + 14 + 16 = 90

, so the mean is

\frac{90 + x}{10}

. Setting the mean equal to the median gives the equation

\frac{90 + x}{10} = 11

. Multiplying both sides by 10 gives

90 + x = 110

, and subtracting 90 yields

x = 20

. This value is consistent with the condition

x > 16

.

Step-by-Step Solution

1

Sort the dataset including the variable

x

.

The sorted dataset of 10 integers is

4, 6, 8, 8, 10, 12, 12, 14, 16, x

.

Since

x > 16

, it must be the largest value in the dataset and will be positioned at the end of the sorted list.

2

Find the median of the sorted dataset.

The median is 11.

For an even number of values (10), the median is the average of the two middle values (the 5th and 6th values), which are 10 and 12:

(10 + 12) / 2 = 11

.

3

Find the sum of the known values and write an expression for the mean.

The mean is

\frac{90 + x}{10}

.

The sum of the 9 known integers is

4 + 6 + 8 + 8 + 10 + 12 + 12 + 14 + 16 = 90

. Adding

x

gives a total sum of

90 + x

, and dividing by the total count of 10 gives the mean.

4

Equate the mean to the median and solve for

x

.

x = 20

Setting the mean expression equal to the median yields

\frac{90 + x}{10} = 11

. Multiplying by 10 gives

90 + x = 110

, which solves to

x = 20

.

Key Concept

Calculating and equating the mean and median of a dataset containing a variable.

Answer

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