All practice questions

Question 2061Question →

In the equation $x^2 - bx + 16 = 0$ , $b$ is a positive integer. If the equation has no real solutions, what is the greatest possible value of $b$ ?

Show answer & explanation

Answer: 7

Answer

The greatest possible value of

b

is

7

.

For the quadratic equation

x^2 - bx + 16 = 0

to have no real solutions, its discriminant must be less than

0

. The discriminant is

(-b)^2 - 4(1)(16) = b^2 - 64

. Solving the inequality

b^2 - 64 < 0

gives

b^2 < 64

. Since

b

is a positive integer, taking the square root of both sides gives

b < 8

. The positive integers less than

8

are

1, 2, 3, 4, 5, 6,

and

7

. The greatest of these values is

7

.

Step-by-Step Solution

1

Set up the inequality for the discriminant to be less than zero.

(-b)^2 - 4(1)(16) < 0

A quadratic equation of the form

ax^2 + bx + c = 0

has no real solutions if and only if its discriminant,

D = b^2 - 4ac

, is negative.

2

Simplify the quadratic inequality.

b^2 < 64

Squaring

-b

yields

b^2

and calculating

4(1)(16)

yields

64

.

3

Solve for the greatest positive integer value of

b

.

b < 8

, so the greatest positive integer is

7

.

Since

b

is a positive integer, the values satisfying

b^2 < 64

are

1, 2, 3, 4, 5, 6,

and

7

. The largest of these is

7

.

Key Concept

Quadratic Discriminant and Number of Solutions

Question 2062Question →

What is the complete set of solutions to the inequality $-\frac{2}{3}(6x - 9) + 4 > 12$ ?

x > -\frac{1}{2}

x < -\frac{1}{2}

x < -\frac{7}{2}

x < -2

Show answer & explanation

Answer: $x < -\frac{1}{2}$

Answer

The inequality is satisfied for all values of

x

such that

x < -\frac{1}{2}

.

Distributing

-\frac{2}{3}

across

(6x - 9)

yields

-4x + 6

. Adding 4 gives

-4x + 10 > 12

. Subtracting 10 from both sides results in

-4x > 2

. Dividing by

-4

and reversing the inequality sign yields

x < -\frac{1}{2}

.

Step-by-Step Solution

1

Distribute

-\frac{2}{3}

to both terms inside the parentheses:

(6x - 9)

.

-4x + 6 + 4 > 12

, which simplifies to

-4x + 10 > 12

.

Applying the distributive property removes the parentheses.

2

Subtract 10 from both sides of the inequality to isolate the term with

x

.

-4x > 2

Subtracting 10 from both sides maintains the inequality while simplifying the constant terms.

3

Divide both sides by

-4

and reverse the inequality sign.

x < -\frac{1}{2}

Dividing both sides of an inequality by a negative number requires reversing the direction of the inequality symbol.

Key Concept

Solving linear inequalities in one variable requires distributing coefficient terms, combining constants, and reversing the inequality sign when multiplying or dividing by a negative number.

Estimated Time:1m 15s

Question 2063Question →

A solid right circular cone has a base radius of $6\text{ centimeters}$ and a height of $8\text{ centimeters}$ . A sphere is inscribed inside the cone such that it is tangent to the base of the cone and tangent to the lateral surface of the cone. What is the ratio of the volume of the sphere to the volume of the cone?

\frac{1}{9}

\frac{3}{8}

\frac{1}{8}

\frac{8}{9}

Show answer & explanation

Answer: $\frac{3}{8}$

Answer

The ratio of the volume of the sphere to the volume of the cone is

\frac{3}{8}

.

The correct answer is found by first determining the radius of the inscribed sphere. By analyzing the 2D cross-section of the cone, which is an isosceles triangle with a base of

12

and a height of

8

, the slant height is found to be

10

using the Pythagorean theorem. The radius of the inscribed circle (the sphere's radius) is calculated using the area and semiperimeter of the triangle, yielding

R = 3

. The volume of the sphere is

36\pi

and the volume of the cone is

96\pi

. Dividing the sphere's volume by the cone's volume gives the ratio of

\frac{3}{8}

.

Step-by-Step Solution

1

Find the slant height of the cone.

Slant height

L = 10\text{ centimeters}

.

The vertical cross-section of the cone is an isosceles triangle with base

2r = 12

and height

h = 8

. The slant height is the hypotenuse of the right triangle formed by the radius, height, and slant height:

L = \sqrt{6^2 + 8^2} = 10

.

2

Calculate the radius of the inscribed sphere.

Sphere radius

R = 3\text{ centimeters}

.

The sphere's radius is the inradius of the cross-sectional isosceles triangle. The area of this triangle is

A = \frac{1}{2} \times 12 \times 8 = 48

, and its semiperimeter is

s = \frac{12 + 10 + 10}{2} = 16

. Using the inradius formula

R = \frac{A}{s}

, we get

R = \frac{48}{16} = 3

.

3

Calculate the volume of the cone.

Volume of the cone

V_{\text{cone}} = 96\pi\text{ cubic centimeters}

.

Using the volume of a cone formula,

V_{\text{cone}} = \frac{1}{3}\pi r^2 h = \frac{1}{3}\pi (6^2)(8) = 96\pi

.

4

Calculate the volume of the sphere.

Volume of the sphere

V_{\text{sphere}} = 36\pi\text{ cubic centimeters}

.

Using the volume of a sphere formula,

V_{\text{sphere}} = \frac{4}{3}\pi R^3 = \frac{4}{3}\pi (3^3) = 36\pi

.

5

Compute the ratio of the volume of the sphere to the volume of the cone.

Ratio

= \frac{3}{8}

.

Divide the volume of the sphere by the volume of the cone:

\frac{36\pi}{96\pi} = \frac{36}{96} = \frac{3}{8}

.

Key Concept

Calculating the volume of spheres and cones, and using 2D cross-sections to find missing geometric dimensions of inscribed solids.

Question 2064Question →

A commercial coffee roaster heats a large batch of coffee beans. The temperature $T$ , in degrees Celsius ( $^\circ\text{C}$ ), of the beans $m$ minutes after heating begins is modeled by the equation $T = 18m + 31$ , where $m \le 10$ . Which of the following is the best interpretation of $18$ in this context?

The increase in the temperature of the beans, in degrees Celsius, for each minute of heating.

The initial temperature of the beans, in degrees Celsius, when heating begins.

The number of minutes it takes for the temperature of the beans to increase by

1^\circ\text{C}

.

The temperature of the beans, in degrees Celsius, after

18

minutes of heating.

Show answer & explanation

Answer: The increase in the temperature of the beans, in degrees Celsius, for each minute of heating.

Answer

The increase in the temperature of the beans, in degrees Celsius, for each minute of heating.

The equation is in the slope-intercept form

T = 18m + 31

, where

18

is the slope and

31

is the y-intercept. In this context,

T

represents the temperature of the beans and

m

represents the time in minutes. The slope,

18

, represents the rate of change of the temperature with respect to time, which means the temperature increases by

18^\circ\text{C}

for each minute of heating.

Step-by-Step Solution

1

Identify the component of the linear equation being questioned.

The number

18

is the coefficient of the variable

m

in the linear equation

T = 18m + 31

.

We need to determine what role the coefficient of the independent variable plays in a linear model.

2

Determine the mathematical meaning of the coefficient in a linear equation of the form

y = mx + b

.

The coefficient of

m

, which is

18

, represents the slope or the rate of change of

T

with respect to

m

.

In a linear function, the slope represents the change in the dependent variable per unit change in the independent variable.

3

Interpret the rate of change in the context of the problem.

Since

T

is the temperature in degrees Celsius (

^\circ\text{C}

) and

m

is the time in minutes, a rate of change of

18

means that the temperature increases by

18^\circ\text{C}

for every

1

minute of heating.

This matches the definition of the slope in the given context.

Key Concept

Interpreting the slope of a linear relationship in context

Estimated Time:1m 0s

Question 2065Question →

If $4^{a - 1} \cdot 8^{2a} = 16^3$ , what is the value of $a$ ?

\\frac{9}{8}

\\frac{13}{8}

\\frac{7}{4}

\\frac{11}{4}

Show answer & explanation

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

Answer

\\frac{7}{4}

The correct answer is

\frac{7}{4}

. Rewriting each base in the equation

4^{a - 1} \cdot 8^{2a} = 16^3

as a power of 2 gives

(2^2)^{a-1} \cdot (2^3)^{2a} = (2^4)^3

. Applying the power of a power rule results in

2^{2a-2} \cdot 2^{6a} = 2^{12}

. Using the product rule of exponents to combine the left side yields

2^{2a-2+6a} = 2^{8a-2} = 2^{12}

. Setting the exponents equal gives

8a - 2 = 12

, which simplifies to

8a = 14

, or

a = \frac{7}{4}

.

Step-by-Step Solution

1

Rewrite each base in the equation

4^{a - 1} \cdot 8^{2a} = 16^3

as a power of 2.

(2^2)^{a-1} \cdot (2^3)^{2a} = (2^4)^3

To solve an exponential equation with different bases, rewrite the bases so they are identical.

2

Apply the power of a power rule

(x^m)^n = x^{m \cdot n}

to simplify each term.

2^{2a-2} \cdot 2^{6a} = 2^{12}

This simplifies the exponents by multiplying the inner and outer exponents.

3

Apply the product rule

x^m \cdot x^n = x^{m+n}

to combine the terms on the left side.

2^{8a-2} = 2^{12}

This combines the exponents of the terms with the common base of 2.

4

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

a

.

8a - 2 = 12 \Rightarrow 8a = 14 \Rightarrow a = \\frac{7}{4}

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

Key Concept

Solving exponential equations by expressing all terms with a common base and applying exponent rules.

Alternative Method

Instead of converting to base 2, all terms can be written in base 4:

4^{a - 1} \cdot (4^{1.5})^{2a} = (4^2)^3

, which simplifies to

4^{a - 1 + 3a} = 4^6

, leading to

4a - 1 = 6

and

a = \\frac{7}{4}

.

Estimated Time:1m 30s

Question 2066Question →

A circular rug has an area of $16\pi$ square feet. If the radius of the rug is tripled, what is the area, in square feet, of the new rug?

32\pi

48\pi

144\pi

64\pi

Show answer & explanation

Answer: $144\pi$

Answer

The area of the new rug is

144\pi

square feet.

The area of a circle scales with the square of its radius. If the radius is tripled, the area is multiplied by

3^2 = 9

. Since the original area is

16\pi

square feet, the area of the new rug is

16\pi \times 9 = 144\pi

square feet.

Step-by-Step Solution

1

Determine the relationship between the linear scale factor and the area scale factor of a circle.

If the radius of a circle is multiplied by a scale factor

k

, the area of the circle is multiplied by

k^2

.

The area of a circle is proportional to the square of its radius (

A = \pi r^2

).

2

Find the area scale factor when the radius is tripled.

Since the radius is multiplied by

3

, the area scale factor is

3^2 = 9

.

Squaring the linear scale factor of 3 gives the quadratic scale factor for the area.

3

Calculate the area of the new rug.

The new area is

16\pi \times 9 = 144\pi

square feet.

Multiplying the original area of

16\pi

square feet by the area scale factor of 9 yields the final scaled area.

Key Concept

Area scaling of two-dimensional shapes

Estimated Time:1m 0s

Question 2067Question →

Two straight support beams on a bridge intersect at a single point. One of the angles formed by their intersection measures $74^\circ$ . An adjacent angle along the straight line of one of the beams has a measure of $(2x + 16)^\circ$ . What is the value of $x$ ?

29

45

61

90

Show answer & explanation

Answer: 45

Answer

45

Since the two angles form a linear pair along a straight support beam, their measures must sum to

180^\circ

. We can write the equation as

74 + (2x + 16) = 180

. Combining like terms gives

2x + 90 = 180

. Subtracting

90

from both sides yields

2x = 90

. Finally, dividing by

2

gives

x = 45

.

Step-by-Step Solution

1

Set up the equation using the property of supplementary angles.

74 + (2x + 16) = 180

Adjacent angles on a straight line form a linear pair, which means they are supplementary and sum to

180^\circ

.

2

Combine the constant terms on the left side of the equation.

2x + 90 = 180

Adding

74

and

16

simplifies the constant terms to

90

.

3

Isolate the variable term by subtracting 90 from both sides of the equation.

2x = 90

Subtracting

90

from

180

leaves

90

on the right side.

4

Solve for x by dividing both sides of the equation by 2.

x = 45

Dividing

2x

and

90

by

2

isolates the variable

x

.

Key Concept

Adjacent angles on a straight line are supplementary and add up to 180 degrees.

Estimated Time:45s

Question 2068Question →

If $27^{x-1} = \sqrt{3^{5x+1}}$ , what is the value of $x$ ?

Show answer & explanation

Answer: 7

Answer

The correct answer is 7.

The correct answer is 7. By converting the base of 27 to

3^3

and rewriting the radical on the right side as a fractional exponent of

\frac{1}{2}

, the equation becomes

3^{3(x-1)} = 3^{\frac{5x+1}{2}}

. Since the bases are equal, their exponents must be equal:

3x - 3 = \frac{5x+1}{2}

. Multiplying both sides by 2 yields

6x - 6 = 5x + 1

. Subtracting

5x

and adding 6 to both sides isolates

x

, giving the solution

x = 7

.

Step-by-Step Solution

1

Express both sides of the equation with a common base of 3.

(3^3)^{x-1} = (3^{5x+1})^{\frac{1}{2}}

Expressing terms with a common base allows the exponents to be compared directly.

2

Apply the power of a power exponent rule,

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

, to simplify the exponents on both sides.

3^{3x-3} = 3^{\frac{5x+1}{2}}

Simplifying the expressions makes it possible to set the exponent expressions equal to each other.

3

Equate the exponents since the bases are identical.

3x - 3 = \frac{5x+1}{2}

If two exponential expressions with the same positive base (other than 1) are equal, their exponents must also be equal.

4

Solve the linear equation for

x

by clearing the fraction and isolating the variable.

x = 7

Multiplying both sides by 2 gives

6x - 6 = 5x + 1

. Subtracting

5x

from both sides and adding 6 to both sides isolates

x

, resulting in

x = 7

.

Key Concept

Solving exponential equations by converting to a common base and applying exponent laws.

Question 2069Question →

During their annual migration across the North Atlantic Gyre, hatchling loggerhead sea turtles rely on an internal compass known as magnetoreception to navigate. The young turtles detect subtle variations in the intensity and angle of Earth's magnetic field ______ sophisticated sensory ability allows them to maintain their direction in the open ocean where visual cues are completely absent.

Which choice completes the text so that it conforms to the conventions of Standard English?

; this

, this

this

, but this

Show answer & explanation

Answer: ; this

Answer

the option containing a semicolon followed by 'this'

The correct option correctly uses a semicolon to join two independent clauses. The clause before the blank ('The young turtles detect subtle variations in the intensity and angle of Earth's magnetic field') and the clause after the blank ('this sophisticated sensory ability allows them to maintain their direction in the open ocean where visual cues are completely absent') are both complete thoughts that can stand alone. A semicolon is a standard and appropriate way to link two closely related independent clauses without a coordinating conjunction.

Step-by-Step Solution

1

Identify the structure of the two clauses surrounding the blank.

The clause before the blank ('The young turtles detect subtle variations in the intensity and angle of Earth's magnetic field') and the clause after the blank ('this sophisticated sensory ability allows them to maintain their direction...') are both independent clauses because each can stand alone as a complete sentence.

Understanding clause boundaries helps determine the correct punctuation or conjunction needed to link them.

2

Evaluate the choices based on standard punctuation rules for linking independent clauses.

Two independent clauses must be joined by a period, a semicolon, a colon (if the second clause explains/illustrates the first), or a comma followed by a coordinating conjunction.

This narrows down the grammatical options and eliminates choices that cause run-on sentences or comma splices.

3

Analyze the semantic relationship between the two clauses to select the most appropriate option.

The second clause explains how the detection mentioned in the first clause helps the turtles navigate. This is a relationship of explanation or addition, not contrast. A semicolon alone is a grammatically correct way to link these closely related independent clauses.

This ensures the selected option is both grammatically correct and logically coherent.

Key Concept

Clause Boundaries and Linking

Estimated Time:1m 0s

Question 2070Question →

An aquarium filter recirculates water at a constant rate of $250$ milliliters per second. The filter cartridge must be replaced after it has processed a total of $540$ cubic meters of water. For how many hours of continuous operation can the filter run before the cartridge must be replaced? ( $1\text{ cubic meter} = 1,000\text{ liters}$ )

Show answer & explanation

Answer: 600

Answer

The filter can run for

600

hours of continuous operation before the cartridge must be replaced.

To find the number of hours the filter can run, we first convert the flow rate to cubic meters per hour. The flow rate of

250

milliliters per second is equal to

0.25

liters per second. Multiplying by

3,600

seconds per hour gives a flow rate of

900

liters per hour. Since

1\text{ cubic meter} = 1,000\text{ liters}

,

900

liters per hour is equivalent to

0.9

cubic meters per hour. Finally, dividing the total capacity of

540

cubic meters by the flow rate of

0.9

cubic meters per hour yields a duration of

600

hours.

Step-by-Step Solution

1

Convert the water recirculating rate from milliliters per second to liters per second.

0.25

liters per second

To express the flow rate in terms of liters before converting to cubic meters.

2

Convert the rate from liters per second to liters per hour.

900

liters per hour

To align the time unit of the rate with the requested time unit (hours).

3

Convert the rate from liters per hour to cubic meters per hour.

0.9

cubic meters per hour

To match the volume unit of the filter cartridge capacity (cubic meters).

4

Divide the total cartridge capacity by the hourly flow rate of the filter.

600

hours

To find the total duration of continuous operation before replacement.

Key Concept

Unit Conversions

Question 2071Question →

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 2072Question →

In right triangle $XYZ$ , the measure of angle $Y$ is $90^\circ$ , $XY = 12$ , and $YZ = 5$ . What is the value of $\cos(X) - \sin(X)$ ?

\frac{7}{13}

\frac{17}{13}

-\frac{7}{13}

\frac{7}{12}

Show answer & explanation

Answer: $\frac{7}{13}$

Answer

\frac{7}{13}

The length of the hypotenuse

XZ

is first found using the Pythagorean theorem:

XZ = \sqrt{12^2 + 5^2} = 13

. Using the definitions of the trigonometric ratios,

\cos(X) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{12}{13}

and

\sin(X) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{5}{13}

. The difference is

\cos(X) - \sin(X) = \frac{12}{13} - \frac{5}{13} = \frac{7}{13}

.

Step-by-Step Solution

1

Find the length of the hypotenuse

XZ

using the Pythagorean theorem.

XZ = \sqrt{XY^2 + YZ^2} = \sqrt{12^2 + 5^2} = \sqrt{144 + 25} = \sqrt{169} = 13

.

The hypotenuse is needed as the denominator for the sine and cosine ratios.

2

Determine the values of

\cos(X)

and

\sin(X)

.

\cos(X) = \frac{XY}{XZ} = \frac{12}{13}

and

\sin(X) = \frac{YZ}{XZ} = \frac{5}{13}

.

By definition,

\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}

and

\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}

.

3

Calculate the difference

\cos(X) - \sin(X)

.

\cos(X) - \sin(X) = \frac{12}{13} - \frac{5}{13} = \frac{7}{13}

.

This is the final subtraction requested by the question.

Key Concept

Calculating trigonometric ratios in a right triangle and applying basic operations.

Question 2073Question →

In the $xy$ -plane, the graph of the quadratic function $f(x) = ax^2 + bx + c$ , where $a$ , $b$ , and $c$ are constants, has a vertex at $(2, -3)$ and passes through the point $(4, 5)$ . If the graph of a second quadratic function, $g$ , is obtained by translating the graph of $f$ horizontally by $3$ units to the right and vertically by $5$ units up, what is the value of $g(5)$ ?

-8

2

8

74

Show answer & explanation

Answer: 2

Answer

The value of

g(5)

is

2

.

The vertex of the graph of

f

is at

(2, -3)

, which means

f(2) = -3

. The graph of

g

is obtained by translating the graph of

f

by

3

units to the right and

5

units up, so its equation is

g(x) = f(x - 3) + 5

. To find

g(5)

, we substitute

x = 5

into this relation, which gives

g(5) = f(5 - 3) + 5 = f(2) + 5

. Since

f(2) = -3

, we have

g(5) = -3 + 5 = 2

. Alternatively, translating the vertex of

f

at

(2, -3)

by

3

units to the right and

5

units up gives the vertex of

g

at

(2 + 3, -3 + 5) = (5, 2)

. Since the vertex of the parabola

g

occurs at

x = 5

, the value of

g(5)

is the

y

-coordinate of the vertex, which is

2

.

Step-by-Step Solution

1

Identify the vertex of the function

f

and write its vertex form equation.

The vertex of

f

is

(2, -3)

, so the vertex form of the function is

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

.

This allows us to find the specific equation of

f(x)

if needed, and also tells us that the value of

f(2)

is

-3

.

2

Set up the equation for the translated function

g(x)

.

A horizontal translation of

3

units to the right and a vertical translation of

5

units up is represented by

g(x) = f(x - 3) + 5

.

This defines the function

g

in terms of the function

f

using standard translation rules.

3

Evaluate

g(5)

using the relation from step 2.

g(5) = f(5 - 3) + 5 = f(2) + 5

.

This simplifies the calculation by using the value of

f

at the vertex.

4

Substitute the value of

f(2)

to find

g(5)

.

g(5) = -3 + 5 = 2

.

Since the vertex of

f

is at

(2, -3)

, we know

f(2) = -3

, which yields the final result.

Key Concept

Quadratic functions can be analyzed and transformed using their vertex form and function translation rules.

Estimated Time:1m 30s

Question 2074Question →

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 2075Question →

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 2076Question →

A group of $25$ students took a history quiz, and their scores are summarized in the frequency table below.

Score	Frequency
6	2
7	4
8	10
9	6
10	3

The teacher decides to remove the scores of the $2$ students who scored $6$ because they were absent on the day of the quiz and took it later under different conditions. Which of the following statements describes the effect of removing these two scores on the mean, median, and standard deviation of the quiz scores?

The mean increases, the median remains unchanged, and the standard deviation decreases.

The mean increases, the median increases, and the standard deviation decreases.

The mean remains unchanged, the median remains unchanged, and the standard deviation increases.

The mean increases, the median remains unchanged, and the standard deviation increases.

Show answer & explanation

Answer: The mean increases, the median remains unchanged, and the standard deviation decreases.

Answer

The mean increases, the median remains unchanged, and the standard deviation decreases.

The correct answer is the statement indicating that the mean increases, the median remains unchanged, and the standard deviation decreases. Removing the two lowest scores (

6

) eliminates the values that pull the average down, resulting in an increased mean. The median remains at

8

because the middle position of the sorted scores still lies within the class of

8

s. The standard deviation decreases because the overall spread of the data is reduced when the lowest and most distant values are removed.

Step-by-Step Solution

1

Calculate the original mean and locate the original median.

Original Mean =

8.16

; Original Median =

8

.

The original sum of scores is

(2 \times 6) + (4 \times 7) + (10 \times 8) + (6 \times 9) + (3 \times 10) = 204

. With

25

students, the mean is

204 / 25 = 8.16

. The median is the

13\text{th}

value in the sorted list of

25

scores. Looking at the cumulative frequencies, the first

2

scores are

6

, the next

4

are

7

(cumulative

6

), and the next

10

are

8

(cumulative

16

). Thus, the

13\text{th}

value is

8

.

2

Calculate the new mean and locate the new median after removing the two scores of

6

.

New Mean

\approx 8.35

; New Median =

8

.

Removing the two scores of

6

reduces the sum by

12

(new sum =

192

) and the count of students to

23

. The new mean is

192 / 23 \approx 8.35

, which is an increase. The new median is the

12\text{th}

value of the remaining

23

sorted scores. The first

4

scores are

7

, and the next

10

are

8

(cumulative

14

). The

12\text{th}

value is still

8

, so the median remains unchanged.

3

Analyze the change in the standard deviation.

The standard deviation decreases.

Standard deviation measures the spread of the data. The scores of

6

are the minimum scores and are the furthest from the mean of the distribution. Removing these extreme values reduces the variability and concentrates the remaining scores closer to the mean, which decreases the standard deviation.

Key Concept

Analyzing the effects of removing extreme data values on the mean, median, and standard deviation of a dataset.

Question 2077Question →

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 2078Question →

A retail store models its weekly profit, $P(d)$ , in dollars, from selling a certain product at a discount of $d$ dollars using the function $P(d) = -5(d - 8)^2 + 2,500$ , where $0 \leq d \leq 20$ . Which of the following is the best interpretation of the value 8 in this context?

The maximum weekly profit, in dollars, that the store can achieve.

The discount, in dollars, that results in the maximum weekly profit.

The weekly profit, in dollars, when no discount is offered.

The discount, in dollars, that results in the minimum weekly profit.

Show answer & explanation

Answer: The discount, in dollars, that results in the maximum weekly profit.

Answer

The discount, in dollars, that results in the maximum weekly profit.

The quadratic function

P(d) = -5(d - 8)^2 + 2,500

is given in vertex form,

P(d) = a(d - h)^2 + k

, where

(h, k)

is the vertex of the parabola. In this equation, the vertex is

(8, 2500)

. The leading coefficient is

-5

, which is negative, meaning the parabola opens downward and the vertex represents the maximum point of the function. In this context,

d

is the discount in dollars and

P(d)

is the weekly profit. Therefore, the value 8 represents the discount of 8 dollars that results in the maximum weekly profit.

Step-by-Step Solution

1

Identify the form of the quadratic function.

The function

P(d) = -5(d - 8)^2 + 2,500

is in vertex form,

P(d) = a(d - h)^2 + k

, where

(h, k)

represents the vertex of the parabola.

Recognizing the vertex form allows us to directly identify the vertex coordinates without expanding the equation.

2

Determine the vertex and the direction the parabola opens.

The vertex is

(8, 2,500)

. Since the leading coefficient

a = -5

is negative, the parabola opens downward, meaning the vertex represents a maximum point.

The sign of the leading coefficient determines whether the vertex represents a maximum or a minimum.

3

Interpret the coordinates of the vertex in context.

The variable

d

is the discount in dollars, and

P(d)

is the weekly profit in dollars. Thus, at the vertex, the discount is 8 dollars, which corresponds to the maximum profit of 2,500 dollars.

Matching the coordinates of the vertex to the variables in the context gives the correct real-world meaning.

Key Concept

Interpreting the vertex of a quadratic function in vertex form within a real-world context.

Estimated Time:1m 0s

Question 2079Question →

A group of $15$ students recorded the number of books they read over the summer. The table below shows the distribution of the number of books read.

Number of books	Number of students
$1$	$1$
$2$	$3$
$3$	$5$
$4$	$5$
$18$	$1$

If the student who read $18$ books is removed from the dataset, which of the following statements best describes the effect on the mean and the median of the dataset?

The mean decreases by

1

, and the median remains the same.

The mean remains the same, and the median decreases by

1

.

Both the mean and the median decrease by

1

.

Both the mean and the median remain the same.

Show answer & explanation

Answer: The mean decreases by $1$ , and the median remains the same.

Answer

The mean decreases by

1

, and the median remains the same.

The correct answer states that the mean decreases by

1

and the median remains the same. Removing the outlier of

18

reduces the sum from

60

to

42

and the number of terms from

15

to

14

, resulting in a new mean of

3

(down from

4

). The median is determined by the middle values of the dataset; both the

8\text{th}

value of the

15

-element set and the average of the

7\text{th}

and

8\text{th}

values of the

14

-element set are

3

, meaning the median does not change.

Step-by-Step Solution

1

Calculate the mean of the original dataset containing

15

students.

The sum of the books read is

(1 \times 1) + (2 \times 3) + (3 \times 5) + (4 \times 5) + (18 \times 1) = 1 + 6 + 15 + 20 + 18 = 60

. The mean is

\frac{60}{15} = 4

.

To find the initial baseline mean before the outlier is removed.

2

Find the median of the original dataset.

With

15

data points, the median is the

8\text{th}

value when ordered. Counting from the smallest value, the first value is

1

, the next

3

values are

2

, and the next

5

values are

3

(covering the

5\text{th}

through

9\text{th}

positions). Thus, the

8\text{th}

value is

3

, so the median is

3

.

To find the initial baseline median before the outlier is removed.

3

Calculate the new mean after removing the student who read

18

books.

The new number of students is

14

. The new sum of books read is

60 - 18 = 42

. The new mean is

\frac{42}{14} = 3

. The mean decreased by

4 - 3 = 1

.

To determine the change in the mean.

4

Find the new median after removing the student who read

18

books.

With

14

data points, the median is the average of the

7\text{th}

and

8\text{th}

values. Both the

7\text{th}

and

8\text{th}

values are in the group of

3

s (covering the

5\text{th}

through

9\text{th}

positions). Therefore, the new median is

\frac{3 + 3}{2} = 3

, meaning the median remains the same.

To determine the change in the median.

Key Concept

Understanding how outliers affect measures of center, specifically the mean (which is sensitive to outliers) and the median (which is resistant to outliers).

Question 2080Question →

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

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