Geometry and Trigonometry Practice Questions — SAT — Page 3

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 42Question →

A solid right circular cone has a base radius of $r$ and a height of $h$ . A right circular cylinder has a base radius that is twice the base radius of the cone, and a height that is three times the height of the cone. What is the ratio of the volume of the cone to the volume of the cylinder?

\frac{1}{36}

\frac{1}{18}

\frac{1}{12}

\frac{1}{6}

Show answer & explanation

Answer: $\frac{1}{36}$

Answer

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

\frac{1}{36}

The volume of a right circular cone is given by

V_{\text{cone}} = \frac{1}{3}\pi r^2 h

. Since the cylinder has a radius that is twice that of the cone (

2r

) and a height that is three times that of the cone (

3h

), its volume is

V_{\text{cylinder}} = \pi (2r)^2(3h) = 12\pi r^2 h

. Dividing the cone's volume by the cylinder's volume gives

\frac{\frac{1}{3}\pi r^2 h}{12\pi r^2 h} = \frac{1}{36}

.

Step-by-Step Solution

1

Write the formula for the volume of the cone.

V_{\text{cone}} = \frac{1}{3}\pi r^2 h

This establishes the volume of the first solid in terms of

r

and

h

.

2

Write the formula for the volume of the cylinder using the scaled dimensions.

V_{\text{cylinder}} = \pi (2r)^2(3h) = 12\pi r^2 h

The cylinder has a radius of

2r

and a height of

3h

, and the volume formula is

\pi R^2 H

.

3

Calculate the ratio of the volume of the cone to the volume of the cylinder.

\frac{V_{\text{cone}}}{V_{\text{cylinder}}} = \frac{\frac{1}{3}\pi r^2 h}{12\pi r^2 h} = \frac{1}{36}

This compares the two volumes directly by dividing the cone's volume by the cylinder's volume.

Key Concept

Volume of cylinders and cones and dimensional scaling relationships.

Alternative Method

Choose convenient sample values for the variables, such as

r = 1

and

h = 3

. The volume of the cone is

V_{\text{cone}} = \frac{1}{3}\pi (1)^2 (3) = \pi

. The cylinder has radius

2(1) = 2

and height

3(3) = 9

, so its volume is

V_{\text{cylinder}} = \pi (2)^2 (9) = 36\pi

. The ratio of the cone's volume to the cylinder's volume is

\frac{\pi}{36\pi} = \frac{1}{36}

.

Estimated Time:1m 15s

Question 43Question →

A solid metal sphere with radius $r$ is melted down and recast into a right circular cylinder with height $\frac{3}{4}r$ . If the total surface area of the cylinder is $k\pi r^2$ , what is the value of $k$ ?

\frac{17}{9}

\frac{7}{2}

\frac{34}{9}

\frac{50}{9}

Show answer & explanation

Answer: $\frac{50}{9}$

Answer

\frac{50}{9}

The volume of the sphere with radius

r

is

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

. When it is melted and recast into a right circular cylinder with radius

R

and height

h = \frac{3}{4}r

, the volume remains the same. The volume of the cylinder is

V = \pi R^2 h

. Equating the two volumes gives

\frac{4}{3}\pi r^3 = \pi R^2 \left(\frac{3}{4}r\right)

. Dividing both sides by

\pi r

yields

\frac{4}{3} r^2 = \frac{3}{4} R^2

, which simplifies to

R^2 = \frac{16}{9} r^2

. Taking the square root of both sides gives

R = \frac{4}{3}r

. The total surface area of the cylinder is given by the formula

S = 2\pi R^2 + 2\pi R h

. Substituting the values of

R

and

h

in terms of

r

yields

S = 2\pi \left(\frac{4}{3}r\right)^2 + 2\pi \left(\frac{4}{3}r\right)\left(\frac{3}{4}r\right) = 2\pi \left(\frac{16}{9}r^2\right) + 2\pi r^2 = \frac{32}{9}\pi r^2 + \frac{18}{9}\pi r^2 = \frac{50}{9}\pi r^2

. Since the surface area is

k\pi r^2

, the value of

k

is

\frac{50}{9}

.

Step-by-Step Solution

1

Express the volume of the sphere in terms of its radius

r

.

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

This is the standard formula for the volume of a sphere.

2

Equate the volume of the sphere to the volume of the cylinder to find the cylinder's radius

R

in terms of

r

.

\frac{4}{3}\pi r^3 = \pi R^2 \left(\frac{3}{4}r\right) \implies R^2 = \frac{16}{9}r^2 \implies R = \frac{4}{3}r

The volume of a cylinder is

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

. Since the sphere is melted and recast into the cylinder, their volumes must be equal. Solving for

R^2

and taking the square root gives the radius

R

.

3

Use the total surface area formula for the cylinder to express the surface area in terms of

r

.

A = 2\pi R^2 + 2\pi R h = 2\pi\left(\frac{16}{9}r^2\right) + 2\pi\left(\frac{4}{3}r\right)\left(\frac{3}{4}r\right) = \frac{32}{9}\pi r^2 + 2\pi r^2 = \frac{50}{9}\pi r^2

The total surface area of a right circular cylinder consists of the area of the two circular bases and the lateral surface area.

4

Compare the calculated surface area to the given expression

k\pi r^2

to find the value of

k

.

k = \frac{50}{9}

Equating

\frac{50}{9}\pi r^2

and

k\pi r^2

yields the constant value

k

.

Key Concept

Equating volumes of solids and calculating total surface area of a cylinder

Estimated Time:2m 30s

Question 44Question →

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 45Question →

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 46Question →

In the triangles $PQR$ and $STU$ shown, $\angle P \cong \angle S$ and $\angle Q \cong \angle T$ . If the length of $PQ$ is $10$ , the length of $QR$ is $15$ , and the length of $ST$ is $6$ , what is the length of $TU$ ?

4

9

11

25

Show answer & explanation

Answer: 9

Answer

9

Since two angles of triangle

PQR

are congruent to two angles of triangle

STU

, the triangles are similar by the Angle-Angle similarity theorem. Therefore, the ratio of corresponding side lengths is constant, which gives the equation

\frac{PQ}{ST} = \frac{QR}{TU}

. Substituting the given values yields

\frac{10}{6} = \frac{15}{TU}

. Solving for

TU

gives

10 \times TU = 90

, which simplifies to

TU = 9

.

Step-by-Step Solution

1

Identify the relationship between the two triangles.

The triangles are similar (

\triangle PQR \sim \triangle STU

).

Since two angles of triangle

PQR

are congruent to two angles of triangle

STU

, the triangles are similar by the Angle-Angle (AA) similarity theorem.

2

Set up a proportion using corresponding side lengths.

\frac{PQ}{ST} = \frac{QR}{TU}

Corresponding sides of similar triangles are proportional.

3

Substitute the known values and solve for the unknown side length.

TU = 9

Substituting the values gives

\frac{10}{6} = \frac{15}{TU}

. Cross-multiplying yields

10 \times TU = 90

, which simplifies to

TU = 9

.

Key Concept

Triangle Similarity (AA Postulate)

Alternative Method

Find the scale factor from triangle

PQR

to triangle

STU

, which is

\frac{ST}{PQ} = \frac{6}{10} = 0.6

. Then, multiply the corresponding side

QR

by this scale factor:

15 \times 0.6 = 9

.

Estimated Time:45s

Question 47Question →

In triangle $XYZ$ , point $W$ lies on side $XY$ and point $V$ lies on side $XZ$ such that line segment $WV$ is parallel to side $YZ$ . If $XW = 4$ , $WY = 2$ , and $WV = 6$ , what is the length of side $YZ$ ?

Show answer & explanation

Answer: 9

Answer

9

Because segment

WV

is parallel to segment

YZ

, triangle

XWV

is similar to triangle

XYZ

by the Angle-Angle (AA) similarity theorem. The ratio of the corresponding side lengths is constant, which gives the proportion

\frac{YZ}{WV} = \frac{XY}{XW}

. The length of side

XY

is

XW + WY = 4 + 2 = 6

. Substituting the values into the proportion yields

\frac{YZ}{6} = \frac{6}{4}

. Solving for

YZ

gives

YZ = 9

.

Step-by-Step Solution

1

Calculate the length of side

XY

XY = 6

The length of side

XY

is the sum of segment lengths

XW

and

WY

:

XY = XW + WY = 4 + 2 = 6

.

2

Determine that triangle

XWV

is similar to triangle

XYZ

\triangle XWV \sim \triangle XYZ

Because segment

WV

is parallel to segment

YZ

, corresponding angles

\angle XWV

and

\angle XYZ

are congruent, and corresponding angles

\angle XVW

and

\angle XZY

are congruent. Since they also share

\angle X

, the two triangles are similar by the Angle-Angle (AA) similarity criterion.

3

Set up a proportion and solve for

YZ

YZ = 9

Corresponding sides of similar triangles are proportional:

\frac{YZ}{WV} = \frac{XY}{XW}

. Substituting the known values gives

\frac{YZ}{6} = \frac{6}{4}

. Multiplying both sides by 6 yields

YZ = \frac{36}{4} = 9

.

Key Concept

Finding side lengths in similar triangles using the Angle-Angle similarity theorem when a line is parallel to one side of a triangle.

Question 48Question →

Triangle $ABC$ is similar to triangle $DEF$ , where vertex $A$ corresponds to vertex $D$ , vertex $B$ corresponds to vertex $E$ , and vertex $C$ corresponds to vertex $F$ . The length of side $AB$ is $6$ , the length of side $BC$ is $8$ , and the length of side $DE$ is $3$ . What is the length of side $EF$ ?

16

4

5

2.25

Show answer & explanation

Answer: $4$

Answer

The correct answer is

4

.

The correct answer is

4

. Since triangle

ABC

is similar to triangle

DEF

, the ratio of their corresponding side lengths must be equal. We set up the proportion

\frac{AB}{DE} = \frac{BC}{EF}

. Substituting the given values gives

\frac{6}{3} = \frac{8}{EF}

, which simplifies to

2 = \frac{8}{EF}

. Solving for

EF

gives

EF = 4

.

Step-by-Step Solution

1

Identify corresponding sides of the two similar triangles.

Since triangle

ABC

is similar to triangle

DEF

, side

AB

corresponds to side

DE

and side

BC

corresponds to side

EF

.

The ordering of vertices in similarity statements determines corresponding parts.

2

Set up a proportion using the ratios of corresponding sides.

The proportion is

\frac{AB}{DE} = \frac{BC}{EF}

, which translates to

\frac{6}{3} = \frac{8}{EF}

.

Corresponding sides of similar triangles are proportional.

3

Solve for the unknown length

EF

.

2 = \frac{8}{EF}

, which gives

EF = \frac{8}{2} = 4

.

Isolate the variable by simplifying the ratio and solving the algebraic equation.

Key Concept

The ratio of corresponding side lengths in similar triangles is constant.

Question 49Question →

A right circular cylinder has a height of $h$ and a base radius of $r$ . A sphere has a radius of $2r$ . If the volume of the sphere is $8$ times the volume of the cylinder, what is the value of the ratio $\frac{h}{r}$ ?

\frac{1}{3}

\frac{3}{4}

\frac{4}{3}

\frac{8}{3}

Show answer & explanation

Answer: $\frac{4}{3}$

Answer

\frac{4}{3}

The volume of a sphere with radius

R

is

\frac{4}{3}\pi R^3

. Substituting

R = 2r

gives a volume of

\frac{4}{3}\pi (2r)^3 = \frac{32}{3}\pi r^3

. The volume of a cylinder is

\pi r^2 h

. Since the volume of the sphere is

8

times the volume of the cylinder, we can set up the equation

\frac{32}{3}\pi r^3 = 8\pi r^2 h

. Dividing both sides of the equation by

8\pi r^2

yields

\frac{4}{3}r = h

, which means the ratio

\frac{h}{r}

is

\frac{4}{3}

.

Step-by-Step Solution

1

Express the volume of both the cylinder and the sphere in terms of

r

and

h

.

The cylinder has volume

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

. The sphere has radius

2r

, so its volume is

V_{\text{sphere}} = \frac{4}{3}\pi(2r)^3 = \frac{32}{3}\pi r^3

.

This establishes formulas for both volumes using the given variables.

2

Set up the equation using the given relation that the volume of the sphere is

8

times the volume of the cylinder.

\frac{32}{3}\pi r^3 = 8\pi r^2 h

This represents the mathematical relationship specified in the problem statement.

3

Solve for the ratio

\frac{h}{r}

by dividing both sides of the equation by

8\pi r^2

.

\frac{32}{3 \times 8} r = h \implies \frac{4}{3}r = h \implies \frac{h}{r} = \frac{4}{3}

This isolates the ratio of the height to the radius to get the final result.

Key Concept

Using the formulas for the volume of a right circular cylinder and a sphere to solve for an unknown dimension ratio.

Question 50Question →

Triangle $LMN$ is similar to triangle $XYZ$ , where vertex $L$ corresponds to vertex $X$ , vertex $M$ corresponds to vertex $Y$ , and vertex $N$ corresponds to vertex $Z$ . The length of side $LM$ is $5$ , the length of side $XY$ is $15$ , and the length of side $YZ$ is $24$ . What is the length of side $MN$ ?

8

72

14

9

Show answer & explanation

Answer: 8

Answer

8

The correct answer is 8. Similar triangles have corresponding side lengths that are proportional. The ratio of the length of side

LM

to the length of its corresponding side

XY

is

5

to

15

, which simplifies to

1

to

3

. Since side

MN

corresponds to side

YZ

, the ratio of the length of side

MN

to the length of side

YZ

must also be

1

to

3

. Setting up the proportion

\frac{1}{3} = \frac{MN}{24}

and solving for the length of side

MN

yields

8

.

Step-by-Step Solution

1

Identify corresponding sides of the similar triangles and set up a proportion.

\frac{LM}{XY} = \frac{MN}{YZ}

Similar triangles have proportional corresponding side lengths.

2

Substitute the given values into the proportion.

\frac{5}{15} = \frac{MN}{24}

The given lengths are

LM = 5

,

XY = 15

, and

YZ = 24

.

3

Simplify the fraction and solve for

MN

.

MN = 8

\frac{5}{15}

simplifies to

\frac{1}{3}

. Multiplying both sides of

\frac{1}{3} = \frac{MN}{24}

by

24

yields

MN = 8

.

Key Concept

Triangle Similarity and Proportional Side Lengths

Alternative Method

Find the scale factor from triangle

LMN

to triangle

XYZ

by dividing

XY

by

LM

:

15 / 5 = 3

. Since triangle

XYZ

is

3

times larger than triangle

LMN

, divide the length of side

YZ

by

3

to find the length of corresponding side

MN

:

24 / 3 = 8

.

Estimated Time:45s

Question 51Question →

In similar triangles $JKL$ and $WXY$ , the ratio of the length of side $JK$ to the length of its corresponding side $WX$ is $1$ to $4$ . If the length of side $WX$ is $x + 9$ and the length of side $JK$ is $x$ , what is the value of $x$ ?

Show answer & explanation

Answer: 3

Answer

The correct answer is 3.

Because triangles

JKL

and

WXY

are similar, the ratio of corresponding side lengths is constant. Since the ratio of

JK

to

WX

is

1

to

4

, the relationship can be written as

\frac{x}{x + 9} = \frac{1}{4}

. Cross-multiplying yields

4x = x + 9

. Subtracting

x

from both sides gives

3x = 9

, and dividing by

3

gives the final answer

x = 3

.

Step-by-Step Solution

1

Set up a proportion using the ratio of corresponding side lengths.

\frac{x}{x + 9} = \frac{1}{4}

The ratio of corresponding side lengths in similar triangles is constant.

2

Cross-multiply to eliminate fractions.

4x = x + 9

Multiplying both sides by both denominators simplifies the equation.

3

Solve for the variable

x

.

x = 3

Subtracting

x

from both sides gives

3x = 9

, and dividing by

3

gives

x = 3

.

Key Concept

The corresponding side lengths of similar triangles are proportional.

Question 52Question →

A solid right circular cylinder has a radius of $r$ centimeters and a height of $6$ centimeters. A solid sphere has a radius of $r$ centimeters. If the sum of the volumes of the cylinder and the sphere is equal to the volume of a right circular cylinder with a radius of $2r$ centimeters and a height of $2$ centimeters, what is the value of $r$ ?

Show answer & explanation

Answer: 1.5

Answer

The correct answer is 1.5 (or 3/2).

The correct answer is 1.5. By using the formulas for the volume of a cylinder (

V = \pi r^2 h

) and a sphere (

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

), the total volume of the first cylinder and the sphere is

6\pi r^2 + \frac{4}{3}\pi r^3

. The second cylinder has a volume of

\pi(2r)^2(2) = 8\pi r^2

. Setting them equal gives

6\pi r^2 + \frac{4}{3}\pi r^3 = 8\pi r^2

. Subtracting

6\pi r^2

from both sides gives

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

. Dividing both sides by

\pi r^2

(since

r > 0

) simplifies to

\frac{4}{3}r = 2

, which solves to

r = 1.5

.

Step-by-Step Solution

1

Express the volume of the first cylinder using the formula

V = \pi r^2 h

.

V_1 = 6\pi r^2

To represent the cylinder's volume in terms of its radius.

2

Express the volume of the sphere using the formula

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

.

V_2 = \frac{4}{3}\pi r^3

To represent the sphere's volume in terms of its radius.

3

Express the volume of the second cylinder, which has a radius of

2r

and a height of

2

.

V_3 = \pi (2r)^2(2) = 8\pi r^2

To represent the larger cylinder's volume in terms of

r

.

4

Set up the equation where the sum of the first cylinder and sphere volumes equals the second cylinder volume.

6\pi r^2 + \frac{4}{3}\pi r^3 = 8\pi r^2

To relate the volumes as given in the problem statement.

5

Simplify the equation by subtracting

6\pi r^2

from both sides.

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

To isolate the cubic term on one side.

6

Divide both sides of the equation by

\pi r^2

since

r > 0

.

\frac{4}{3}r = 2

To reduce the polynomial equation to a linear equation.

7

Multiply both sides by

\frac{3}{4}

to solve for

r

.

r = 1.5

To find the final numerical value of the radius.

Key Concept

Volume of cylinders and spheres

Question 53Question →

A rectangular prism has a length of $8$ centimeters, a width of $6$ centimeters, and a height of $h$ centimeters. A right circular cylinder has a base radius of $3$ centimeters and a height of $h$ centimeters. If the volume of the rectangular prism is $V$ cubic centimeters and the volume of the cylinder is $W$ cubic centimeters, which of the following expressions represents the ratio of $V$ to $W$ ?

\frac{16}{3\pi}

\frac{16}{3}

\frac{3\pi}{16}

\frac{16}{\pi}

Show answer & explanation

Answer: $\frac{16}{3\pi}$

Answer

The ratio of the volume of the rectangular prism to the volume of the cylinder is

\frac{16}{3\pi}

.

The volume of the rectangular prism is the product of its dimensions:

V = 8 \times 6 \times h = 48h

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

\pi r^2 h

:

W = \pi \times 3^2 \times h = 9\pi h

. The ratio of

V

to

W

is represented by the fraction

\frac{V}{W} = \frac{48h}{9\pi h}

. Simplifying the fraction by canceling the common variable

h

and dividing the numerator and denominator by

3

yields the expression

\frac{16}{3\pi}

.

Step-by-Step Solution

1

Write the expression for the volume of the rectangular prism,

V

, in terms of

h

.

V = 8 \times 6 \times h = 48h

The volume of a rectangular prism is calculated using the formula

\text{Volume} = lwh

, where

l

is the length,

w

is the width, and

h

is the height.

2

Write the expression for the volume of the cylinder,

W

, in terms of

h

.

W = \pi \times 3^2 \times h = 9\pi h

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

\text{Volume} = \pi r^2 h

, where

r

is the base radius and

h

is the height.

3

Express the ratio of

V

to

W

as a fraction and simplify it.

\frac{V}{W} = \frac{48h}{9\pi h} = \frac{48}{9\pi} = \frac{16}{3\pi}

The ratio of

V

to

W

is given by

\frac{V}{W}

. The common height variable

h

cancels out from both the numerator and the denominator, and the fraction

\frac{48}{9}

simplifies to

\frac{16}{3}

by dividing both numbers by their greatest common divisor,

3

.

Key Concept

Calculating and comparing volumes of rectangular prisms and right circular cylinders in terms of algebraic variables.

Question 54Question →

In the $xy$ -plane, a square has vertices at $(0,0)$ , $(10,0)$ , $(10,10)$ , and $(0,10)$ . A line that passes through the points $(0,2)$ and $(8,10)$ divides the square into two regions. What is the area of the larger region?

Show answer & explanation

Answer: 68

Answer

The area of the larger region is 68.

The total area of the square is

10^2 = 100

. The line segment between

(0,2)

on the left boundary and

(8,10)

on the top boundary forms a right triangle with the top-left vertex of the square

(0,10)

. The legs of this right triangle have lengths

10 - 2 = 8

and

8 - 0 = 8

, so its area is

\frac{1}{2} \times 8 \times 8 = 32

. The area of the other region is

100 - 32 = 68

. The larger area is therefore 68.

Step-by-Step Solution

1

Calculate the area of the square

100

The square has vertices at

(0,0)

,

(10,0)

,

(10,10)

, and

(0,10)

, which gives a side length of 10. The area of a square is

\text{side}^2

.

2

Determine the dimensions of the smaller triangular region formed by the line

Legs of length 8 and 8

The line intersects the left edge at

(0,2)

and the top edge at

(8,10)

. The corner of the square is at

(0,10)

. The distance from

(0,2)

to

(0,10)

is 8, and the distance from

(8,10)

to

(0,10)

is 8.

3

Calculate the area of the smaller triangular region

32

The area of a right triangle is

\frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 8 \times 8 = 32

.

4

Subtract the area of the smaller region from the total area of the square to find the area of the larger region

68

The area of the larger region is the total area of the square minus the area of the smaller region:

100 - 32 = 68

.

Key Concept

Calculating the area of a region by partitioning a geometric shape or using subtraction of areas.

Question 55Question →

In right triangle $ABC$ , the measure of angle $B$ is $90^\circ$ . If the length of side $AB$ is 6 and the length of side $AC$ is 10, what is the length of side $BC$ ?

Show answer & explanation

Answer: 8

Answer

The length of side

BC

is 8.

By applying the Pythagorean theorem to right triangle

ABC

, we have

AB^2 + BC^2 = AC^2

. Substituting the given values gives

6^2 + BC^2 = 10^2

, or

36 + BC^2 = 100

. Subtracting 36 from both sides gives

BC^2 = 64

, and taking the square root yields

BC = 8

.

Step-by-Step Solution

1

Identify the hypotenuse and the given leg in the right triangle.

The hypotenuse is

AC = 10

and the given leg is

AB = 6

.

The side opposite the

90^\circ

angle (angle

B

) is always the hypotenuse.

2

Apply the Pythagorean theorem.

AB^2 + BC^2 = AC^2

, which becomes

6^2 + BC^2 = 10^2

.

The Pythagorean theorem states that the sum of the squares of the legs of a right triangle equals the square of the hypotenuse.

3

Solve for the unknown leg

BC

.

36 + BC^2 = 100 \implies BC^2 = 64 \implies BC = 8

.

Simplify the squares, subtract 36 from both sides, and find the principal square root of 64.

Key Concept

The Pythagorean Theorem

Question 56Question →

A right circular cylinder and a right circular cone have equal volumes. The radius of the cylinder's base is $3$ times the radius of the cone's base. If the height of the cone is $k$ times the height of the cylinder, what is the value of $k$ ?

Show answer & explanation

Answer: 27

Answer

27

The volume of a cylinder is

V_{\text{cylinder}} = \pi r_{\text{cylinder}}^2 h_{\text{cylinder}}

and the volume of a cone is

V_{\text{cone}} = \frac{1}{3} \pi r_{\text{cone}}^2 h_{\text{cone}}

. Let the base radius of the cone be

r

and the height of the cylinder be

h

. According to the problem, the base radius of the cylinder is

3r

and the height of the cone is

kh

. Setting the volumes equal to each other gives

\pi (3r)^2 h = \frac{1}{3} \pi r^2 (kh)

. Squaring the term in parentheses simplifies the equation to

9 \pi r^2 h = \frac{k}{3} \pi r^2 h

. Dividing both sides by the common term

\pi r^2 h

yields

9 = \frac{k}{3}

. Multiplying both sides by 3 results in

k = 27

.

Step-by-Step Solution

1

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

V_{\text{cylinder}} = \pi r_{\text{cylinder}}^2 h_{\text{cylinder}}

and

V_{\text{cone}} = \frac{1}{3} \pi r_{\text{cone}}^2 h_{\text{cone}}

To establish the mathematical equations governing the volume of each solid.

2

Define variables for the cone's radius (

r

) and the cylinder's height (

h

), then write the cylinder's radius and the cone's height using the given relationships.

r_{\text{cylinder}} = 3r

,

r_{\text{cone}} = r

,

h_{\text{cylinder}} = h

, and

h_{\text{cone}} = kh

To express all variables in terms of

r

,

h

, and the constant

k

so they can be compared directly.

3

Substitute the expressions into the volume formulas and set the two volumes equal to each other.

\pi (3r)^2 h = \frac{1}{3} \pi r^2 (kh)

The problem states that the cylinder and the cone have equal volumes.

4

Simplify the equation by squaring the cylinder's radius and dividing both sides by the common factors.

9 \pi r^2 h = \frac{k}{3} \pi r^2 h \implies 9 = \frac{k}{3} \implies k = 27

To isolate and solve for the constant

k

.

Key Concept

Relating the volumes of geometric solids through algebraic substitution and dimension scaling.

Question 57Question →

A landscape architect is designing two similar gardens. The smaller garden is in the shape of a regular hexagon and has an area of $24\sqrt{3}$ square meters. The larger garden is also a regular hexagon, and its perimeter is $3$ times the perimeter of the smaller garden. An inscribed circular fountain is to be placed in the center of the larger garden, touching all six sides. What is the area, in square meters, of the circular fountain?

12\pi

36\pi

108\pi

144\pi

Show answer & explanation

Answer: $108\pi$

Answer

The area of the circular fountain is

108\pi

square meters.

The side length of the smaller hexagon is calculated to be

4

meters using the area formula for a regular hexagon. Because the larger hexagon's perimeter is scaled by a factor of

3

, its side length is

12

meters. The radius of the inscribed circle is the apothem of the larger hexagon, which is

12 \times \frac{\sqrt{3}}{2} = 6\sqrt{3}

meters. Squaring this radius and multiplying by

\pi

results in an area of

108\pi

square meters.

Step-by-Step Solution

1

Find the side length of the smaller regular hexagon.

The side length is

4

meters.

The area of a regular hexagon with side length

s_1

is given by

A = \frac{3\sqrt{3}}{2} s_1^2

. Setting this equal to

24\sqrt{3}

yields

s_1^2 = 16

, so

s_1 = 4

.

2

Determine the side length of the larger regular hexagon.

The side length is

12

meters.

Since the two hexagons are similar, the ratio of their perimeters is equal to the linear scale factor of their sides. The perimeter of the larger hexagon is

3

times that of the smaller hexagon, so its side length is

s_2 = 3 \times 4 = 12

.

3

Find the radius of the inscribed circle in the larger hexagon.

The radius is

6\sqrt{3}

meters.

The radius

r

of the inscribed circle is the apothem of the regular hexagon. For a hexagon with side length

s_2 = 12

, the apothem is

r = s_2 \frac{\sqrt{3}}{2} = 12 \frac{\sqrt{3}}{2} = 6\sqrt{3}

.

4

Calculate the area of the inscribed circular fountain.

The area is

108\pi

square meters.

Using the area formula for a circle,

A = \pi r^2

, we substitute

r = 6\sqrt{3}

to get

A = \pi (6\sqrt{3})^2 = 108\pi

.

Key Concept

Area of regular polygons, similarity ratio scaling, and properties of inscribed circles.

Question 58Question →

A rectangular garden has a length of $12$ feet and a width of $5$ feet. A gardener wants to place a straight wooden diagonal walk from one corner of the garden to the opposite corner. What is the length, in feet, of the diagonal walk?

17

\sqrt{119}

13

8.5

Show answer & explanation

Answer: $13$

Answer

The length of the diagonal walk is

13

feet.

The diagonal of the rectangular garden forms a right triangle with the garden's length and width acting as the legs. By applying the Pythagorean theorem,

a^2 + b^2 = c^2

, where

a = 5

and

b = 12

, we get

5^2 + 12^2 = 25 + 144 = 169

. Taking the square root of

169

gives

13

. Thus, the length of the diagonal walk is

13

feet.

Step-by-Step Solution

1

Identify the right triangle formed by the length, width, and diagonal of the rectangle.

A right triangle is formed where the two legs are the length (

12

feet) and the width (

5

feet), and the hypotenuse is the diagonal walk (

d

).

The corners of a rectangle form

90^\circ

angles, making the diagonal the hypotenuse of a right triangle.

2

Apply the Pythagorean theorem,

a^2 + b^2 = c^2

, where

a = 5

,

b = 12

, and

c = d

.

5^2 + 12^2 = d^2

To relate the legs of a right triangle to its hypotenuse.

3

Simplify the equation to solve for

d

.

25 + 144 = d^2

, which simplifies to

169 = d^2

. Taking the square root of both sides gives

d = 13

.

To find the numerical value of the hypotenuse.

Key Concept

Right Triangles and the Pythagorean Theorem

Estimated Time:45s

Question 59Question →

An acute angle $\theta$ in a right triangle satisfies $\cos(\theta) = \frac{3}{5}$ . If $\phi$ is the other acute angle of the triangle, what is the value of $\sin(\phi) \tan(\theta)$ ?

\frac{16}{15}

\frac{9}{20}

\frac{4}{5}

\frac{12}{25}

Show answer & explanation

Answer: $\frac{4}{5}$

Answer

four-fifths

The correct answer is four-fifths. Because

\theta

and

\phi

are complementary angles in a right triangle,

\sin(\phi)

equals

\cos(\theta)

, which is given as three-fifths. Using the Pythagorean identity,

\sin(\theta)

is four-fifths, making

\tan(\theta)

equal to four-thirds. Multiplying

\sin(\phi)

and

\tan(\theta)

gives three-fifths times four-thirds, which simplifies to four-fifths.

Step-by-Step Solution

1

Use the complementary angle relationship to find

\sin(\phi)

.

\sin(\phi) = \cos(\theta) = \frac{3}{5}

The two acute angles in a right triangle,

\theta

and

\phi

, sum to

90^\circ

. Therefore, the sine of one angle equals the cosine of the other.

2

Find

\sin(\theta)

using the Pythagorean identity.

\sin(\theta) = \frac{4}{5}

Since

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

, we have

\sin^2(\theta) + \left(\frac{3}{5}\right)^2 = 1

, which gives

\sin^2(\theta) = \frac{16}{25}

. Because

\theta

is acute,

\sin(\theta) = \frac{4}{5}

.

3

Calculate

\tan(\theta)

using the quotient identity.

\tan(\theta) = \frac{4}{3}

By definition,

\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} = \frac{4/5}{3/5} = \frac{4}{3}

.

4

Multiply

\sin(\phi)

by

\tan(\theta)

.

\sin(\phi) \tan(\theta) = \frac{4}{5}

Substitute the found values:

\sin(\phi) \tan(\theta) = \frac{3}{5} \times \frac{4}{3} = \frac{4}{5}

.

Key Concept

Trigonometric co-function identities and quotient identities in right triangles

Question 60Question →

A straight ladder is leaning against a vertical wall. The base of the ladder is placed $9$ feet from the bottom of the wall. If the top of the ladder touches the wall at a height of $12$ feet above the ground, what is the length, in feet, of the ladder?

Show answer & explanation

Answer: 15

Answer

The length of the ladder is

15

feet.

By representing the scenario as a right triangle, the two perpendicular sides (legs) have lengths of

9

feet and

12

feet. Using the Pythagorean theorem (

a^2 + b^2 = c^2

), we find

9^2 + 12^2 = 81 + 144 = 225

. Taking the square root of

225

gives the hypotenuse length, which is

15

feet.

Step-by-Step Solution

1

Model the scenario using a right triangle.

A right triangle where the vertical leg is

12

feet (height on the wall), the horizontal leg is

9

feet (distance along the ground), and the hypotenuse is

c

(the ladder length).

The intersection of the vertical wall and the horizontal ground forms a right angle (

90^\circ

).

2

Set up the Pythagorean equation.

9^2 + 12^2 = c^2

The Pythagorean theorem states that the sum of the squares of the legs of a right triangle equals the square of the hypotenuse.

3

Solve for the hypotenuse

c

.

c = \sqrt{81 + 144} = \sqrt{225} = 15

Simplifying the arithmetic yields

225

, and taking the square root gives the final length of

15

feet.

Key Concept

Pythagorean Theorem

Estimated Time:45s

Geometry and Trigonometry

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