All practice questions

If $\sin(x^\circ) = \cos(38^\circ)$ , where $0 < x < 90$ , what is the value of $x$ ?

Show answer & explanation

Answer: 52

Answer

The value of

x

is 52.

Applying the co-function identity

\sin(\theta) = \cos(90^\circ - \theta)

allows us to set the argument of the sine function as the complement of the cosine's argument. Thus,

x = 90 - 38 = 52

.

Step-by-Step Solution

1

Identify the relevant trigonometric identity.

The co-function identity

\sin(x^\circ) = \cos(90^\circ - x^\circ)

is appropriate here.

We need to relate the sine of one angle to the cosine of another angle.

2

Equate the angle arguments using the identity.

\sin(x^\circ) = \cos(90^\circ - x^\circ) = \cos(38^\circ)

, which implies

90 - x = 38

.

For acute angles, if their cosine values are equal, the angles themselves must be equal.

3

Solve the linear equation for

x

.

x = 90 - 38 = 52

.

Isolating

x

gives the final answer.

Key Concept

Co-function identities relate the sine of an angle to the cosine of its complement:

\sin(\theta) = \cos(90^\circ - \theta)

.

Question 462Question →

A triangular banner has an area of $30$ square feet. If the height of the banner is $5$ feet, what is the length, in feet, of the base of the banner?

Show answer & explanation

Answer: 12

Answer

The length of the base of the banner is

12

feet.

To find the base of the triangular banner, apply the formula for the area of a triangle:

A = \frac{1}{2} b h

. Substituting

30

for the area

A

and

5

for the height

h

yields

30 = \frac{1}{2} b (5)

, or

30 = 2.5b

. Dividing both sides of the equation by

2.5

gives

b = 12

. Therefore, the length of the base of the banner is

12

feet.

Step-by-Step Solution

1

Recall the formula for the area of a triangle.

A = \frac{1}{2} b h

The area of a triangle is equal to half the product of its base and height.

2

Substitute the given values into the area formula.

30 = \frac{1}{2} \times b \times 5

The problem provides the area (

30

square feet) and the height (

5

feet).

3

Solve for the base

b

.

b = 12

Multiply both sides of the equation by

2

to clear the fraction, giving

60 = 5b

. Then, divide both sides by

5

to find that

b = 12

.

Key Concept

Area of a triangle

Question 463Question →

A scientist measures the density of a core sample of ice as a function of depth. The density $d$ , in grams per cubic centimeter ( $\text{g/cm}^3$ ), of the ice at a depth of $x$ meters below the glacier surface is modeled by the equation $d = 0.0004x + 0.917$ . According to the model, what is the depth, in meters, at which the ice density is $0.935\text{ g/cm}^3$ ?

Show answer & explanation

Answer: 45

Answer

The depth is 45 meters.

To find the depth at which the density is

0.935\text{ g/cm}^3

, substitute

0.935

for the density

d

in the model equation, yielding

0.935 = 0.0004x + 0.917

. Subtracting

0.917

from both sides gives

0.018 = 0.0004x

. Dividing both sides by

0.0004

results in

x = 45

. Therefore, the depth is 45 meters.

Step-by-Step Solution

1

Substitute the target density of

0.935\text{ g/cm}^3

for

d

in the given model equation.

0.935 = 0.0004x + 0.917

This sets up the linear equation to solve for the corresponding depth

x

.

2

Subtract

0.917

from both sides of the equation.

0.018 = 0.0004x

This isolates the term containing the variable

x

on one side of the equation.

3

Divide both sides of the equation by

0.0004

to find the value of

x

.

x = 45

Dividing

0.018

by

0.0004

solves for the depth

x

in meters.

Key Concept

Solving linear relationships in context for the independent variable given a value of the dependent variable.

Question 464Question →

A laboratory uses two types of equipment, Instrument A and Instrument B, which both have an initial value of $V$ dollars when purchased at the same time ( $t = 0$ years). The value of Instrument A decreases linearly at a constant rate over time. At $t = 3$ years, the value of Instrument A is $\$12,000$ , and at $t = 6$ years, its value is $\$8,000$ . The value of Instrument B decreases exponentially at a constant annual rate. At $t = 2$ years, the value of Instrument B is $\$4,000$ . What is the value, in dollars, of Instrument B at $t = 4$ years?

Show answer & explanation

Answer: 1000

Answer

1000

The correct answer is 1000. By defining the value of Instrument A linearly, we obtain the system

V - 3d = 12000

and

V - 6d = 8000

. Solving this system yields an initial value of

V = 16000

. Instrument B decreases exponentially with the model

V_B(t) = 16000 \cdot b^t

. Given

V_B(2) = 4000

, we have

16000 \cdot b^2 = 4000

, which gives

b^2 = 0.25

. The value at

t = 4

is then calculated as

V_B(4) = V_B(2) \cdot b^2 = 4000 \cdot 0.25 = 1000

.

Step-by-Step Solution

1

Set up a system of linear equations for Instrument A's value.

V - 3d = 12000

and

V - 6d = 8000

Instrument A decreases linearly, meaning its value is modeled by

V_A(t) = V - dt

.

2

Solve for the constant annual rate of decrease,

d

, and the initial purchase value,

V

.

d = \frac{4000}{3}

and

V = 16000

Subtracting the two linear equations eliminates

V

, allowing us to solve for

d

and subsequently substitute it back to find

V

.

3

Apply the initial value

V = 16000

to Instrument B's exponential decay model at

t = 2

.

16000 \cdot b^2 = 4000

Instrument B decreases exponentially starting from the same initial value

V

, so its model is

V_B(t) = V \cdot b^t

.

4

Solve for the 2-year decay factor,

b^2

.

b^2 = 0.25

Dividing both sides of the equation by

16000

isolates the term representing the growth or decay factor over a two-year interval.

5

Calculate the value of Instrument B at

t = 4

years.

1000

Multiply the value at

t = 2

by the 2-year decay factor

b^2

, which yields

4000 \cdot 0.25 = 1000

.

Key Concept

Distinguishing between linear decay (constant absolute rate of change) and exponential decay (constant percentage rate of change) to set up and solve coupled system models.

Question 465Question →

If $\sin(\theta) = \frac{\sqrt{5}}{3}$ and $\theta$ is an acute angle, what is the value of $\tan^2(\theta)$ ?

Show answer & explanation

Answer: 1.25

Answer

The correct answer is 1.25 (or the equivalent fraction 5/4).

Using the Pythagorean identity

\sin^2(\theta) + \cos^2(\theta) = 1

, we find

\cos^2(\theta) = 1 - \frac{5}{9} = \frac{4}{9}

. Since

\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}

, it follows that

\tan^2(\theta) = \frac{\sin^2(\theta)}{\cos^2(\theta)} = \frac{5/9}{4/9} = 1.25

.

Step-by-Step Solution

1

Calculate the square of the cosine of the angle using the Pythagorean identity.

\cos^2(\theta) = 1 - \sin^2(\theta) = 1 - \left(\frac{\sqrt{5}}{3}\right)^2 = 1 - \frac{5}{9} = \frac{4}{9}

The Pythagorean identity states that

\sin^2(\theta) + \cos^2(\theta) = 1

for any angle

\theta

.

2

Express the tangent squared in terms of sine squared and cosine squared.

\tan^2(\theta) = \frac{\sin^2(\theta)}{\cos^2(\theta)}

By definition, the tangent function is the ratio of sine to cosine, so

\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}

.

3

Substitute the values of

\sin^2(\theta)

and

\cos^2(\theta)

to solve for

\tan^2(\theta)

.

\tan^2(\theta) = \frac{5/9}{4/9} = \frac{5}{4} = 1.25

Plugging in the squared ratios and simplifying gives the final evaluation.

Key Concept

Pythagorean trigonometric identity and quotient identity of tangent

Question 466Question →

For the function $f$ , selected values of $x$ and $f(x)$ are shown in the table below.

$x$	$f(x)$
$-1$	$4$
$1$	$2$
$3$	$-1$
$5$	$6$

The function $g$ is defined by $g(x) = a \cdot f(x - 2) + 5$ , where $a$ is a constant. If $g(5) = 3$ , what is the value of $a$ ?

Show answer & explanation

Answer: 2

Answer

2

Evaluating

g(5)

using the formula

g(x) = a \cdot f(x - 2) + 5

requires finding

f(5 - 2) = f(3)

. From the table,

f(3) = -1

. Substituting these values yields

3 = a(-1) + 5

, which simplifies to

-a = -2

, or

a = 2

.

Step-by-Step Solution

1

Express

g(5)

using the given definition of

g(x)

.

g(5) = a \cdot f(3) + 5

By substituting

x = 5

into the definition

g(x) = a \cdot f(x - 2) + 5

, we obtain

g(5) = a \cdot f(5 - 2) + 5 = a \cdot f(3) + 5

.

2

Find the value of

f(3)

from the table.

f(3) = -1

Looking at the row where

x = 3

in the table, the corresponding output

f(x)

is

-1

.

3

Substitute the known values into the equation for

g(5)

and solve for

a

.

a = 2

Substitute

g(5) = 3

and

f(3) = -1

into the equation to get

3 = a(-1) + 5

. Subtracting 5 from both sides gives

-2 = -a

, which simplifies to

a = 2

.

Key Concept

Evaluating a transformed function using a table of values.

Question 467Question →

The graph of the quadratic function $f(x) = x^2 - 6x + c$ , where $c$ is a constant, has its vertex at $(h, k)$ in the $xy$ -plane. If the graph of $f$ is translated $3$ units to the right and $2$ units down, the vertex of the translated graph lies on the line $y = 2x$ . What is the value of $c$ ?

Show answer & explanation

Answer: 23

Answer

23

To find the constant

c

, we first determine the vertex of the function

f(x) = x^2 - 6x + c

. The x-coordinate of the vertex of a parabola in the form

y = ax^2 + bx + c

is given by

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

. For this function,

x = -\frac{-6}{2(1)} = 3

. Substituting this back into the function gives the y-coordinate of the vertex:

f(3) = (3)^2 - 6(3) + c = c - 9

. Thus, the original vertex is at

(3, c - 9)

. Translating the graph

3

units to the right increases the x-coordinate of the vertex by

3

, making it

3 + 3 = 6

. Translating the graph

2

units down decreases the y-coordinate of the vertex by

2

, making it

(c - 9) - 2 = c - 11

. The problem states that this new vertex

(6, c - 11)

lies on the line

y = 2x

. Substituting these coordinates into the linear equation gives

c - 11 = 2(6)

, which simplifies to

c - 11 = 12

. Adding

11

to both sides gives the value of

c

as

23

.

Step-by-Step Solution

1

Find the vertex

(h, k)

of the original quadratic function

f(x) = x^2 - 6x + c

.

The vertex is at

(3, c - 9)

.

The x-coordinate of the vertex of a quadratic function

y = ax^2 + bx + c

is given by

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

. For

f(x) = x^2 - 6x + c

, we have

h = -\frac{-6}{2(1)} = 3

. Substituting

x = 3

into the function gives the y-coordinate:

k = f(3) = 3^2 - 6(3) + c = c - 9

.

2

Determine the coordinates of the vertex after translating the graph

3

units to the right and

2

units down.

The new vertex is at

(6, c - 11)

.

A translation of

3

units to the right increases the x-coordinate of the vertex by

3

, so the new x-coordinate is

3 + 3 = 6

. A translation of

2

units down decreases the y-coordinate of the vertex by

2

, so the new y-coordinate is

(c - 9) - 2 = c - 11

.

3

Set up an equation using the line

y = 2x

and solve for

c

.

c = 23

Since the translated vertex

(6, c - 11)

lies on the line

y = 2x

, substituting

x = 6

and

y = c - 11

into the line's equation must satisfy it:

c - 11 = 2(6) \Rightarrow c - 11 = 12 \Rightarrow c = 23

.

Key Concept

Determining the vertex of a quadratic function and applying translations to its graph.

Estimated Time:1m 30s

Question 468Question →

In the inequality $-3(2x - 5) + 8 \geq 2(kx - 3) - 1$ , $k$ is a constant. If the solution set for $x$ is $x \leq 2$ , what is the value of $k$ ?

Show answer & explanation

Answer: 4.5

Answer

4.5

To find the value of

k

, simplify both sides of the inequality:

-3(2x - 5) + 8 \geq 2(kx - 3) - 1

becomes

-6x + 23 \geq 2kx - 7

. Grouping the variable terms on the left and constants on the right gives

-(6 + 2k)x \geq -30

. Because the solution set is

x \leq 2

, dividing by the negative coefficient reverses the inequality sign to yield

x \leq \frac{30}{6 + 2k}

. Setting the boundary value

\frac{30}{6 + 2k}

equal to 2 gives the equation

30 = 2(6 + 2k) = 12 + 4k

. Subtracting 12 from both sides gives

18 = 4k

, which results in

k = 4.5

.

Step-by-Step Solution

1

Simplify both sides of the inequality

-6x + 23 \geq 2kx - 7

Distribute the constants on both sides:

-3(2x - 5) + 8 = -6x + 15 + 8 = -6x + 23

, and

2(kx - 3) - 1 = 2kx - 6 - 1 = 2kx - 7

.

2

Isolate the variable terms on the left side and constants on the right side

-(6 + 2k)x \geq -30

Subtract

2kx

and

23

from both sides to get

-6x - 2kx \geq -7 - 23

, then factor out

x

to get

-(6 + 2k)x \geq -30

.

3

Relate the inequality to the given solution boundary

x \leq \frac{30}{6 + 2k}

Since the solution is

x \leq 2

, dividing both sides by the negative coefficient

-(6 + 2k)

reverses the inequality sign, yielding

x \leq \frac{-30}{-(6 + 2k)} = \frac{30}{6 + 2k}

.

4

Solve for

k

using the boundary equation

k = 4.5

Set the boundary expression equal to 2:

\frac{30}{6 + 2k} = 2

. Multiply both sides by

6 + 2k

to get

30 = 12 + 4k

, which simplifies to

18 = 4k

, giving

k = 4.5

.

Key Concept

Solving linear inequalities in one variable involving variable coefficients and applying the inequality direction flip when dividing by a negative value.

Question 469Question →

In the quadratic equation $x^2 - 6x - k = 0$ , $k$ is a positive constant. If the solutions to the equation are $x = 3 \pm \sqrt{17}$ , what is the value of $k$ ?

Show answer & explanation

Answer: 8

Answer

8

By applying the quadratic formula to

x^2 - 6x - k = 0

, we find the solutions are

x = 3 \pm \sqrt{9 + k}

. Equating the expression inside the radical to the given solutions

3 \pm \sqrt{17}

yields

9 + k = 17

. Solving for

k

gives 8.

Step-by-Step Solution

1

Identify the coefficients of the quadratic equation.

a = 1

,

b = -6

, and

c = -k

To apply the quadratic formula, we need to know the values of

a

,

b

, and

c

from the standard form

ax^2 + bx + c = 0

.

2

Substitute the coefficients into the quadratic formula.

x = \frac{6 \pm \sqrt{(-6)^2 - 4(1)(-k)}}{2(1)}

The quadratic formula

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

gives the solutions directly.

3

Simplify the expression inside the radical and divide by 2.

x = 3 \pm \sqrt{9 + k}

Simplifying the radical expression allows us to easily compare it to the given solutions.

4

Compare the simplified solution to the given solution form.

9 + k = 17

Since the solutions are given as

3 \pm \sqrt{17}

, the expression inside the radical must equal 17.

5

Solve for the constant

k

.

k = 8

Subtracting 9 from both sides of the equation yields the value of the constant.

Key Concept

Solving quadratic equations using the quadratic formula or by completing the square to find the relationship between coefficients and solutions.

Alternative Method

Instead of using the quadratic formula, the equation can be solved by completing the square. Rewrite the equation as

x^2 - 6x = k

. Adding 9 to both sides gives

x^2 - 6x + 9 = k + 9

, which can be factored as

(x - 3)^2 = k + 9

. Taking the square root of both sides yields

x = 3 \pm \sqrt{k + 9}

. Comparing this to the given solutions

3 \pm \sqrt{17}

, we get

k + 9 = 17

, so

k = 8

.

Estimated Time:1m 30s

Question 470Question →

In right triangle $JKL$ , the measure of angle $K$ is $90^\circ$ . If $\cos(J) = 0.28$ , what is the value of $\sin(L)$ ?

Show answer & explanation

Answer: 0.28

Answer

The correct answer is 0.28 (or the equivalent fraction 7/25).

In a right triangle, the two acute angles are complementary, meaning they add up to

90^\circ

. The co-function identity states that the sine of an acute angle is equal to the cosine of its complement. Therefore, in right triangle

JKL

with right angle

K

,

\sin(L) = \cos(J)

. Given that

\cos(J) = 0.28

, the value of

\sin(L)

must also be

0.28

.

Step-by-Step Solution

1

Determine the relationship between angles

J

and

L

.

J + L = 90^\circ

Since the sum of angles in a triangle is

180^\circ

and angle

K

is

90^\circ

, the sum of the remaining two angles must be

180^\circ - 90^\circ = 90^\circ

.

2

Use the co-function trigonometric identity.

\sin(L) = \cos(J)

The sine of an acute angle in a right triangle is equal to the cosine of its complementary angle.

3

Substitute the given value of

\cos(J)

into the identity.

\sin(L) = 0.28

We are given that

\cos(J) = 0.28

.

Key Concept

Complementary angle trigonometric identity (co-function identity)

Question 471Question →

A rectangular box has a square base with side length $x$ inches and a height of $5$ inches. The total surface area of the box is $192$ square inches. What is the value of $x$ ?

Show answer & explanation

Answer: 6

Answer

The value of

x

is

6

.

The total surface area is found by adding the areas of all six faces. The top and bottom faces are squares with side length

x

, so their combined area is

2x^2

. The four vertical sides are rectangles with dimensions

x

by

5

, so their combined area is

4(5x) = 20x

. Setting their sum equal to the total surface area gives

2x^2 + 20x = 192

. Dividing by

2

yields

x^2 + 10x - 96 = 0

, which factors as

(x + 16)(x - 6) = 0

. Since

x

must be positive,

x = 6

.

Step-by-Step Solution

1

Set up the equation for the total surface area of the rectangular box.

\text{Total Surface Area} = 2x^2 + 20x = 192

The total surface area of a rectangular box with a square base of side length

x

and height

h

consists of two square bases (top and bottom) of area

x^2

each, and four rectangular sides of area

xh

each. Substituting

h = 5

gives

2x^2 + 4(5x) = 2x^2 + 20x

.

2

Simplify and set the quadratic equation to zero.

x^2 + 10x - 96 = 0

Dividing all terms by

2

simplifies the equation to

x^2 + 10x = 96

. Subtracting

96

from both sides sets the quadratic equation to standard form

ax^2 + bx + c = 0

.

3

Factor the quadratic equation and solve for the positive value of

x

.

x = 6

Factoring the equation gives

(x + 16)(x - 6) = 0

, which yields solutions of

x = -16

and

x = 6

. Since the side length of a geometric solid must be positive, we reject the negative solution.

Key Concept

Calculating the surface area of a rectangular prism and solving the resulting quadratic relationship.

Estimated Time:1m 35s

Question 472Question →

A marine biologist records the dive depths, in meters, of a seal on 6 consecutive dives. The recorded depths are 52, 68, 57, 65, 48, and $x$ . The median of these 6 dive depths is 58 meters. What is the value of $x$ ?

Show answer & explanation

Answer: 59

Answer

59

To find the median of a set of 6 numbers, the numbers must be listed in ascending order, and the median will be the average of the 3rd and 4th numbers. The five known numbers in ascending order are 48, 52, 57, 65, and 68. If the unknown value

x

is less than or equal to 57, the 3rd and 4th numbers in the sorted list would be at most 57, which would make the median at most 57. If

x

is greater than or equal to 65, the 3rd and 4th numbers in the sorted list would be 57 and 65, which would make the median

\frac{57+65}{2} = 61

. Since the median is given as 58,

x

must be between 57 and 65. Thus, when the 6 numbers are sorted, the 3rd number is 57 and the 4th number is

x

. Setting their average equal to 58 gives

\frac{57+x}{2} = 58

. Multiplying both sides by 2 yields

57 + x = 116

. Subtracting 57 from both sides gives

x = 59

.

Step-by-Step Solution

1

Order the five known depths from least to greatest.

48, 52, 57, 65, 68

To find or use the median of a dataset, the data points must first be arranged in ascending order.

2

Analyze the position of the median in a dataset of 6 values.

The median is the average of the 3rd and 4th values in the ordered list.

For an even number of data points, the median is the arithmetic mean of the two middle values.

3

Determine the constraints on

x

to achieve a median of 58.

The value of

x

must be between 57 and 65, making 57 the 3rd value and

x

the 4th value.

If

x \le 57

, the 3rd and 4th values are both 57 or less, yielding a median of at most 57. If

x \ge 65

, the 3rd and 4th values are 57 and 65, yielding a median of 61. Thus,

x

must be the 4th value.

4

Set up and solve the equation for the median.

\frac{57 + x}{2} = 58 \implies 57 + x = 116 \implies x = 59

Since the 3rd value is 57 and the 4th value is

x

, their average must equal the given median of 58.

Key Concept

Calculating and interpreting the median of a dataset with an even number of values, including handling unknown variables.

Question 473Question →

A scientist surveyed $15$ different forest regions and recorded the number of endangered plant species in each region. The numbers of endangered species in $14$ of the regions are:

12, 15, 15, 16, 18, 20, 20, 22, 22, 22, 25, 28, 30, 34

The number of endangered species in the $15\text{th}$ region is $x$ , where $x$ is an integer greater than $25$ . If the median of the numbers of endangered species for all $15$ regions is equal to the mean, what is the value of $x$ ?

Show answer & explanation

Answer: 31

Answer

The correct answer is 31.

The correct answer is 31. The sum of the 14 known values is 299. Since the 15th value

x

is greater than 25, it must be placed at or after the 12th position when all 15 values are sorted. Thus, the 8th value (the median of the 15 values) remains the same as the 8th value of the known 14 sorted values, which is 22. Setting the mean equal to the median gives

\frac{299 + x}{15} = 22

, which simplifies to

299 + x = 330

, resulting in

x = 31

. Since

31 > 25

, this is the correct value.

Step-by-Step Solution

1

Find the sum of the 14 known values.

The sum of the 14 known values is 299.

This is needed to construct the expression for the mean of the 15 values.

2

Determine the median of the 15 values given the constraint

x > 25

.

The median is 22.

Since

x > 25

, it will be sorted after the first 11 values (the 11th value is 25). The median of 15 values is the 8th value, which remains 22.

3

Set the mean equal to the median and solve for

x

.

\frac{299 + x}{15} = 22 \implies 299 + x = 330 \implies x = 31

.

This yields the value of

x

that satisfies the condition that the mean equals the median.

Key Concept

Calculating and comparing the mean and median of a dataset, and analyzing how a new data point affects these measures.

Question 474Question →

A truck rental company charges a daily fee of $45.00 plus$ 0.75 per mile driven. A driver rents a truck for one day, has a budget of at most $150.00, and is required to drive at least 60 miles for a delivery. What is the maximum number of additional miles the driver can drive beyond the required 60 miles without exceeding the budget?

Show answer & explanation

Answer: 80

Answer

80

To find the maximum number of additional miles, we set up the inequality

45 + 0.75(60 + a) \leq 150

, where

a

represents the number of additional miles driven. Distributing

0.75

yields

45 + 45 + 0.75a \leq 150

, which simplifies to

90 + 0.75a \leq 150

. Subtracting

90

from both sides gives

0.75a \leq 60

. Finally, dividing by

0.75

gives

a \leq 80

. The maximum value of

a

is therefore 80.

Step-by-Step Solution

1

Set up the inequality representing the total budget constraint.

45 + 0.75(60 + a) \leq 150

, where

a

is the number of additional miles.

The daily fee is

45, the per-mile rate is

0.75, the driver must drive at least 60 miles plus

a

additional miles, and the total cost cannot exceed the $150 budget.

2

Simplify the expression by distributing

0.75

and combining constant terms.

90 + 0.75a \leq 150

0.75 \times 60 = 45

, and adding the daily fee of

45

gives

90

.

3

Isolate the variable term by subtracting

90

from both sides.

0.75a \leq 60

This determines the remaining budget available for the additional miles.

4

Solve for

a

by dividing both sides by

0.75

.

a \leq 80

Dividing

60

by

0.75

gives the maximum number of additional miles.

Key Concept

Solving multi-step linear inequalities in context

Question 475Question →

In a certain board game, a player receives $15$ points for each quest completed, but loses $4$ points for each penalty card drawn. A player completes $12$ quests and draws $c$ penalty cards. If the player's total score is greater than $120$ points, what is the maximum possible value of $c$ ?

Show answer & explanation

Answer: 14

Answer

14

The player earns a base score of

15 \times 12 = 180

points from quests, and loses

4

points for each of the

c

penalty cards, resulting in a total score of

180 - 4c

points. Because the score must be greater than

120

, we write the inequality

180 - 4c > 120

. Subtracting

180

from both sides yields

-4c > -60

. Dividing by

-4

and reversing the inequality sign gives

c < 15

. Since

c

must represent a whole number of cards, the maximum possible value is the largest integer less than

15

, which is

14

.

Step-by-Step Solution

1

Set up the inequality representing the score constraint.

180 - 4c > 120

The player earns

15

points for each of the

12

quests (

15 \times 12 = 180

) and loses

4

points for each of the

c

penalty cards (

4c

), and this total must exceed

120

.

2

Isolate the variable term by subtracting

180

from both sides.

-4c > -60

To solve for

c

, we first subtract the constant term

180

from both sides of the inequality.

3

Divide both sides by

-4

and reverse the inequality sign.

c < 15

Dividing an inequality by a negative number requires reversing the direction of the inequality sign.

4

Determine the maximum integer value for

c

.

14

Since the number of penalty cards

c

must be a whole number and

c

must be strictly less than

15

, the largest possible value is

14

.

Key Concept

Solving linear inequalities in one variable involving multiplication or division by a negative number and interpreting the solution set in a discrete context.

Question 476Question →

A botanist is studying the transpiration rate of a certain plant species under controlled conditions. The transpiration rate $T$ , in grams of water vapor per square meter of leaf area per hour ( $\text{g/m}^2/\text{h}$ ), is modeled by a linear function of the relative humidity $H$ , expressed as a decimal where $0.20 \leq H \leq 0.80$ . The model is represented by the equation $T = -4.8H + C$ , where $C$ is a constant. Based on this model, if the relative humidity increases by $0.15$ , what is the corresponding decrease in the transpiration rate, in $\text{g/m}^2/\text{h}$ ?

Show answer & explanation

Answer: 0.72

Answer

The correct answer is 0.72.

In the linear equation

T = -4.8H + C

, the coefficient of

H

(which is

-4.8

) represents the slope of the line. The slope indicates that for every increase of

1.0

in the relative humidity

H

, the transpiration rate

T

decreases by

4.8

grams of water vapor per square meter of leaf area per hour. To find the decrease in transpiration rate corresponding to an increase of

0.15

in relative humidity, multiply the rate of change by the change in relative humidity:

4.8 \times 0.15 = 0.72

. Thus, the transpiration rate decreases by

0.72

grams of water vapor per square meter of leaf area per hour.

Step-by-Step Solution

1

Identify the slope of the linear model.

The slope is

-4.8

.

The coefficient of

H

in the linear equation

T = -4.8H + C

represents the rate of change of the transpiration rate with respect to the relative humidity.

2

Calculate the change in the transpiration rate (

\Delta T

) for an increase of

0.15

in the relative humidity (

\Delta H = 0.15

).

\Delta T = -4.8 \times 0.15 = -0.72

The change in the dependent variable is equal to the slope multiplied by the change in the independent variable.

3

Determine the magnitude of the decrease.

The decrease in the transpiration rate is

0.72

.

The negative sign in the change

\Delta T = -0.72

represents a decrease, so the amount of decrease is

0.72

.

Key Concept

Interpreting the slope of a linear relationship in context as the rate of change of the dependent variable with respect to the independent variable.

Question 477Question →

(2x - 5)(2x + 7) = -k

In the given equation, $k$ is a constant. If the equation has exactly one real solution, what is the value of $k$ ?

Show answer & explanation

Answer: 36

Answer

36

Expanding the equation

(2x - 5)(2x + 7) = -k

gives

4x^2 + 4x - 35 = -k

. Adding

k

to both sides puts it in standard form

4x^2 + 4x + (k - 35) = 0

. For a quadratic equation to have exactly one real solution, its discriminant must be zero:

b^2 - 4ac = 4^2 - 4(4)(k - 35) = 0

. Simplifying this yields

16 - 16(k - 35) = 0

, which gives

1 - (k - 35) = 0

, so

k - 35 = 1

, and thus

k = 36

.

Step-by-Step Solution

1

Expand the left side of the equation and write it in standard form.

4x^2 + 4x + (k - 35) = 0

To apply the quadratic discriminant, the equation must be in the form

ax^2 + bx + c = 0

.

2

Identify the coefficients and set the discriminant

b^2 - 4ac

equal to 0.

4^2 - 4(4)(k - 35) = 0

A quadratic equation has exactly one real solution if and only if its discriminant is equal to zero.

3

Solve the resulting linear equation for the constant

k

.

k = 36

Isolating

k

gives the value that satisfies the condition of having exactly one real solution.

Key Concept

Quadratic Discriminant

Question 478Question →

A shipping service charges a flat rate of $15.50 to ship a box, plus$ 0.50 per ounce of the total weight of the box. A customer wants to ship a package and wants the total shipping cost to be at most $40.00. If the empty box weighs 14 ounces, what is the maximum weight, in ounces, of the contents that can be placed inside the box?

Show answer & explanation

Answer: 35

Answer

The maximum weight of the contents that can be placed inside the box is 35 ounces.

To find the maximum weight of the contents, let

w

represent the weight of the contents in ounces. The total weight of the package is the sum of the content weight and the empty box weight, or

w + 14

ounces. The total shipping cost is the flat rate of

15.50 plus

0.50 times the total weight, which is represented by the expression

15.50 + 0.50(w + 14)

. Since the customer wants the cost to be at most

40.00, we write the inequality

15.50 + 0.50(w + 14) \leq 40.00

. Distributing

0.50

yields

15.50 + 0.50w + 7 \leq 40.00

. Combining the constant terms gives

22.50 + 0.50w \leq 40.00

. Subtracting

22.50

from both sides results in

0.50w \leq 17.50

. Finally, dividing both sides by

0.50

gives

w \leq 35$. Thus, the maximum weight of the contents is 35 ounces.

Step-by-Step Solution

1

Define the variable and write the inequality representing the total cost constraints.

15.50 + 0.50(w + 14) \leq 40.00

The total weight is the sum of the contents (

w

) and the empty box (14 ounces), and the cost is

15.50 plus

0.50 per ounce, which cannot exceed $40.00.

2

Distribute the rate of $0.50 per ounce across the weight terms and simplify.

22.50 + 0.50w \leq 40.00

Applying the distributive property gives

0.50w + 7.00

, and adding the flat rate

15.50

simplifies the left side.

3

Isolate the variable term by subtracting

22.50

from both sides.

0.50w \leq 17.50

This isolates the variable term

0.50w

on one side of the inequality.

4

Solve for

w

by dividing both sides by

0.50

.

w \leq 35

Dividing by

0.50

(or multiplying by 2) yields the maximum weight limit for the contents.

Key Concept

Solving linear inequalities in one variable using distributive properties and combining like terms.

Question 479Question →

The graph of the quadratic function $f$ in the $xy$ -plane has its vertex at $(4, 12)$ . The function is defined by $f(x) = -(x - c)(x - d)$ , where $c$ and $d$ are constants. What is the value of the product $cd$ ?

Show answer & explanation

Answer: 4

Answer

The value of the product

cd

is

4

.

The

x

-coordinate of the vertex of a quadratic function of the form

f(x) = -(x-c)(x-d)

is the average of the

x

-intercepts

c

and

d

. Since the vertex is

(4, 12)

, we have

\frac{c+d}{2} = 4

, which means

c+d = 8

. Substituting the vertex

(4, 12)

into the function gives

12 = -(4-c)(4-d)

, which expands to

12 = -(16 - 4(c+d) + cd)

. Substituting

c+d = 8

gives

12 = -(16 - 32 + cd) = -(-16 + cd) = 16 - cd

. Solving for

cd

yields

cd = 4

.

Step-by-Step Solution

1

Find the sum of the constants

c

and

d

using the

x

-coordinate of the vertex.

c + d = 8

The graph of a quadratic function in the form

f(x) = -(x-c)(x-d)

has a vertical line of symmetry at the

x

-coordinate of its vertex, which is the midpoint of its

x

-intercepts

c

and

d

. Therefore,

\frac{c+d}{2} = 4

, which simplifies to

c + d = 8

.

2

Substitute the vertex coordinates

(4, 12)

into the function definition.

(4-c)(4-d) = -12

Since

(4, 12)

is the vertex, the point lies on the graph of

f

, meaning

f(4) = 12

. Substituting

x = 4

into the function gives

12 = -(4-c)(4-d)

, which simplifies to

(4-c)(4-d) = -12

.

3

Expand the expression

(4-c)(4-d)

and substitute

c+d = 8

to solve for

cd

.

cd = 4

Expanding

(4-c)(4-d) = -12

gives

16 - 4(c+d) + cd = -12

. Substituting

c+d = 8

yields

16 - 32 + cd = -12

, which simplifies to

-16 + cd = -12

. Adding

16

to both sides gives

cd = 4

.

Key Concept

Quadratic Functions and Graphs

Question 480Question →

An agricultural irrigation system draws water from a storage tank at a constant rate. The volume of water, $V$ , in gallons, remaining in the tank $m$ minutes after the irrigation begins is given by the equation $V = 12,000 - 45m$ . According to the model, after how many minutes of irrigation will there be exactly $8,400$ gallons of water remaining in the tank?

Show answer & explanation

Answer: 80

Answer

The irrigation system will have exactly

8,400

gallons of water remaining after

80

minutes.

To find the number of minutes,

m

, when the volume of water remaining is exactly

8,400

gallons, substitute

8,400

for

V

in the given linear model:

8,400 = 12,000 - 45m

. Subtracting

12,000

from both sides of the equation yields

-3,600 = -45m

. Dividing both sides of the equation by

-45

yields

m = 80

. Therefore, after

80

minutes, there will be exactly

8,400

gallons of water remaining in the tank.

Step-by-Step Solution

1

Substitute the target volume of

8,400

gallons for

V

in the linear model equation.

8,400 = 12,000 - 45m

We want to find the number of minutes,

m

, when the remaining volume,

V

, is exactly

8,400

gallons.

2

Subtract

12,000

from both sides of the equation.

-3,600 = -45m

This isolates the variable term

-45m

on one side of the equation.

3

Divide both sides by

-45

to solve for

m

.

m = 80

This isolates

m

to find the number of minutes elapsed.

Key Concept

Solving linear equations in context

Answer

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