Geometry Practice Questions — Armed Services Vocational Aptitude Battery (ASVAB) — Page 7

Question 121Question →

A circular track has an inner circumference of $440$ meters. The track has a uniform width of $7$ meters. What is the circumference of the outer edge of the track? (Use $\pi \approx \frac{22}{7}$ )

447 meters

462 meters

484 meters

528 meters

Show answer & explanation

Answer: 484 meters

Answer

484 meters

The inner circumference is

440

meters. Using the formula

C = 2\pi r

, the inner radius is

70

meters (

440 = 2 \times \frac{22}{7} \times r

). Because the track has a uniform width of

7

meters, the outer radius is

70 + 7 = 77

meters. Using the circumference formula again for the outer edge,

C = 2 \times \frac{22}{7} \times 77 = 484

meters.

Step-by-Step Solution

1

Find the inner radius using the given circumference.

r_{inner} = 70

meters

Using the formula

C = 2\pi r

, substitute

440

for

C

and

\frac{22}{7}

for

\pi

:

440 = 2 \times \frac{22}{7} \times r

, which simplifies to

r = 70

.

2

Calculate the outer radius by adding the track's width.

r_{outer} = 77

meters

The outer edge is

7

meters further from the center than the inner edge, so you must add the width to the inner radius (

70 + 7 = 77

).

3

Calculate the outer circumference.

484

meters

Use the formula

C = 2\pi r

with the new outer radius:

C = 2 \times \frac{22}{7} \times 77 = 44 \times 11 = 484

.

Key Concept

Calculating the circumference of concentric circles by determining and adjusting their respective radii.

Alternative Method

You can also solve this using the diameter. The inner diameter is

140

meters (

440 = \pi d

). Since the track is

7

meters wide on all sides, the total width added to the overall diameter is

14

meters (

7 \times 2

). The outer diameter is

140 + 14 = 154

meters. The outer circumference is

\frac{22}{7} \times 154 = 484

meters.

Estimated Time:1m 30s

Question 122Question →

In a blueprint for a corner bracket, a laser cuts an exact $90^\circ$ right angle, splitting it into two adjacent sections. The measure of one section is determined by the expression $(2x + 18)^\circ$ , while the adjacent section is measured as $(3x - 8)^\circ$ . Based on this design, what is the degree measure of the larger section?

50^\circ

94^\circ

56^\circ

30^\circ

Show answer & explanation

Answer: $50^\circ$

Answer

The larger section measures

50^\circ

.

The problem states the laser cuts a 90-degree right angle into two sections, meaning the two expressions must add up to 90 degrees. Setting up the equation (2x + 18) + (3x - 8) = 90 simplifies to 5x + 10 = 90. Solving for x gives x = 16. Substituting x back into the expressions yields 50 degrees for the first section and 40 degrees for the second section. The larger of the two is 50 degrees.

Step-by-Step Solution

1

Set up the algebraic equation based on the geometric relationship.

(2x + 18) + (3x - 8) = 90

The two adjacent sections form a right angle, meaning they are complementary and must sum to 90 degrees.

2

Combine like terms on the left side of the equation.

5x + 10 = 90

Combining the variable terms and constant terms simplifies the equation.

3

Isolate the variable x.

x = 16

Subtract 10 from both sides to get 5x = 80, then divide by 5.

4

Substitute the value of x back into both angle expressions.

First section =

2(16) + 18 = 50^\circ

; Second section =

3(16) - 8 = 40^\circ

The specific degree measures must be calculated to find the larger angle.

5

Compare the two angle measures.

50^\circ

The question asks specifically for the measure of the larger section.

Key Concept

Complementary Angles and Algebraic Substitution

Question 123Question →

Match each circle-related calculation on the left to its correct numerical value on the right.

Click a left item, then click its matching right item

Show answer & explanation

Answer

Area with radius 5 matches to 25π; Circumference with diameter 10 matches to 10π; Radius of area 100π matches to 10; Diameter of circumference 12π matches to 12.

Each match is verified by accurately applying the formulas

A = \pi r^2

and

C = \pi d

to either solve for the resulting property or to work backwards to find the missing dimension.

Step-by-Step Solution

1

Calculate the area for the first description.

Area =

25\pi

The formula for the area of a circle is

A = \pi r^2

. Substituting

r = 5

yields

25\pi

.

2

Calculate the circumference for the second description.

Circumference =

10\pi

The formula for circumference is

C = \pi d

. Substituting

d = 10

yields

10\pi

.

3

Solve for the radius in the third description.

Radius =

10

Given area

A = 100\pi

, use

100\pi = \pi r^2

to find that

r^2 = 100

, so

r = 10

.

4

Solve for the diameter in the fourth description.

Diameter =

12

Given circumference

C = 12\pi

, use

12\pi = \pi d

to determine that

d = 12

.

Key Concept

Calculating area, circumference, radius, and diameter of a circle using fundamental geometry formulas.

Question 124Question →

A civil engineer is designing a railroad crossing where a straight diagonal track intersects two parallel roads. The two same-side interior angles between the parallel roads and the track are represented by the algebraic expressions $(4x - 20)^\circ$ and $(3x + 25)^\circ$ . Based on the geometric properties of parallel lines intersected by a transversal, match each description on the left with its correct calculated numerical value on the right.

Click a left item, then click its matching right item

Show answer & explanation

Answer

The correct matches are: the value of x is 25, the acute angle is 80°, the obtuse angle is 100°, and the difference between them is 20°.

Same-side interior angles formed by parallel lines and a transversal are supplementary, meaning they add up to 180 degrees. Setting up the equation (4x - 20) + (3x + 25) = 180 allows us to solve for x, which equals 25. Substituting x = 25 back into the original expressions provides the individual angle measures: 80 degrees (acute) and 100 degrees (obtuse). Finally, subtracting the acute angle from the obtuse angle yields a difference of 20 degrees.

Step-by-Step Solution

1

Set up an equation based on the geometric relationship of the angles.

(4x - 20) + (3x + 25) = 180

When a transversal intersects parallel lines, same-side interior angles are supplementary, meaning their sum is exactly 180 degrees.

2

Solve the algebraic equation to find the value of x.

7x + 5 = 180 \rightarrow 7x = 175 \rightarrow x = 25

Combining the like terms and isolating x provides the necessary variable value to determine the actual angle measures.

3

Substitute the value of x back into the original expressions.

4(25) - 20 = 80^\circ

and

3(25) + 25 = 100^\circ

This calculation determines the specific measures of the acute and obtuse angles formed at the intersection.

4

Calculate the difference between the obtuse and acute angle measures.

100^\circ - 80^\circ = 20^\circ

This satisfies the final required numerical value for the matching pairs.

Key Concept

Properties of parallel lines intersected by a transversal, specifically that same-side interior angles are supplementary.

Question 125Question →

A drone is tethered to a fixed landing pad and can fly within a maximum circular boundary with an area of $400\pi$ square meters. If a safety protocol requires the tether to be shortened by $8$ meters, what is the maximum area of the new circular flight boundary?

144\pi \text{ square meters}

256\pi \text{ square meters}

336\pi \text{ square meters}

24\pi \text{ square meters}

Show answer & explanation

Answer: $144\pi \text{ square meters}$

Answer

The new flight boundary has an area of

144\pi \text{ square meters}

.

The initial area of

400\pi

square meters indicates the original tether length (radius) was

20

meters, because

\pi r^2 = 400\pi

. Shortening the tether by

8

meters results in a new radius of

12

meters. The new area is calculated by squaring the new radius and multiplying by

\pi

, yielding

\pi(12)^2 = 144\pi

square meters.

Step-by-Step Solution

1

Find the original radius using the area formula

A = \pi r^2

.

400\pi = \pi r^2

, so

r^2 = 400

, which means

r = 20

meters.

The original tether length represents the radius of the circular boundary.

2

Calculate the new radius by subtracting

8

meters.

20 - 8 = 12

meters.

The safety protocol shortens the tether, directly reducing the flight radius.

3

Calculate the area of the new circular boundary.

A = \pi(12)^2 = 144\pi \text{ square meters}

.

The problem asks for the maximum area of the new boundary.

Key Concept

Area of a circle and its relationship to the radius.

Question 126Question →

An isosceles triangle has two equal sides that each measure $13$ inches and a base that measures $10$ inches. What is the length of the altitude (height) drawn to the base?

8

inches

36

inches

12

inches

18

inches

Show answer & explanation

Answer: $12$ inches

Answer

The length of the altitude is

12

inches.

Drawing an altitude to the base of an isosceles triangle splits it into two identical right triangles. The base of

10

inches is bisected into two

5

-inch segments. For one of these right triangles, the base leg is

5

inches and the hypotenuse (the equal side) is

13

inches. Using the Pythagorean theorem (

a^2 + b^2 = c^2

), we set up the equation

h^2 + 5^2 = 13^2

. Solving for

h^2

gives

169 - 25 = 144

. Taking the square root of

144

yields an altitude of

12

inches.

Step-by-Step Solution

1

Identify the properties of an altitude in an isosceles triangle.

The altitude bisects the

10

-inch base, creating two equal segments of

5

inches each.

This splits the isosceles triangle into two congruent right triangles, allowing the use of the Pythagorean theorem.

2

Set up the Pythagorean theorem (

a^2 + b^2 = c^2

) for one of the right triangles.

h^2 + 5^2 = 13^2

, where

h

is the altitude,

5

is the base leg, and

13

is the hypotenuse.

The equal side of the isosceles triangle serves as the hypotenuse of the newly formed right triangle.

3

Square the known values.

h^2 + 25 = 169

.

To begin isolating the unknown altitude squared (

h^2

).

4

Solve for the altitude.

h^2 = 169 - 25 = 144

, so

h = \sqrt{144} = 12

.

Taking the square root of

144

gives the final length of the altitude.

Key Concept

Using the Pythagorean theorem to find the altitude of an isosceles triangle

Question 127Question →

A flat metal washer is manufactured by stamping a circular hole with a diameter of $4 \text{ mm}$ out of the exact center of a solid circular disk with a diameter of $12 \text{ mm}$ . What is the area of the remaining flat face of the washer in square millimeters? (Enter only the numerical coefficient of $\pi$ in your answer. For example, if the area is $50\pi$ , enter 50).

Show answer & explanation

Answer: 32

Answer

32

The area of the remaining metal washer face is found by calculating the area of the large outer circle (

36\pi

) and subtracting the area of the smaller inner circle hole (

4\pi

), which leaves a remaining area of

32\pi

. Since the question asks for the coefficient of

\pi

, the correct numerical entry is 32.

Step-by-Step Solution

1

Calculate the radius of the outer disk.

R = 12 \div 2 = 6 \text{ mm}

The area formula for a circle requires the radius, which is half of the diameter.

2

Calculate the area of the outer disk.

\text{Area}_{\text{outer}} = \pi \times 6^2 = 36\pi \text{ mm}^2

This represents the total surface area before the center hole is removed.

3

Calculate the radius of the inner hole.

r = 4 \div 2 = 2 \text{ mm}

The radius of the removed section is needed to determine how much area to subtract.

4

Calculate the area of the inner hole.

\text{Area}_{\text{inner}} = \pi \times 2^2 = 4\pi \text{ mm}^2

This is the area of the metal that is stamped out.

5

Subtract the inner area from the outer area.

36\pi - 4\pi = 32\pi \text{ mm}^2

The remaining face area of the washer is the difference between the full disk and the removed hole.

6

Identify the coefficient of

\pi

.

32

The stem specifically asks for the numerical coefficient only.

Key Concept

Calculating the area of an annulus (the region between two concentric circles) by subtracting the area of the inner circle from the area of the outer circle.

Question 128Question →

A searchlight beam sweeps horizontally across a central angle of $72^\circ$ . If the maximum effective range of the searchlight is $20 \text{ meters}$ , what is the length of the arc formed by the outer edge of the beam? (Leave your answer in terms of $\pi$ ).

4\pi \text{ meters}

8\pi \text{ meters}

40\pi \text{ meters}

80\pi \text{ meters}

Show answer & explanation

Answer: $8\pi \text{ meters}$

Answer

8\pi \text{ meters}

The length of an arc is calculated by multiplying the full circumference of the circle (

2\pi r

) by the fraction of the circle represented by the central angle (

\frac{\theta}{360^\circ}

). In this scenario, the full circumference is

2\pi(20) = 40\pi \text{ meters}

. The fraction representing the

72^\circ

sweep is

\frac{72}{360}

, which simplifies to

\frac{1}{5}

. Multiplying the full circumference by this fraction gives

\frac{1}{5} \times 40\pi = 8\pi \text{ meters}

.

Step-by-Step Solution

1

Identify the formula for the arc length of a circle.

\text{Arc Length} = (\frac{\theta}{360^\circ}) \times 2\pi r

The arc length represents a specific fraction of the entire circumference based on the central angle.

2

Determine the fraction of the circle covered by the searchlight beam.

\frac{72^\circ}{360^\circ} = \frac{1}{5}

Simplifying the angle fraction makes the subsequent multiplication easier.

3

Calculate the full circumference of the circle using the given radius (range).

C = 2\pi(20) = 40\pi \text{ meters}

The full circumference is necessary to find the length of the fractional arc.

4

Multiply the full circumference by the angle fraction.

\frac{1}{5} \times 40\pi = 8\pi \text{ meters}

This calculation applies the proportional fraction to the total boundary of the circle, yielding the arc length.

Key Concept

Arc Length of a Circle

Estimated Time:1m 0s

Question 129Question →

Two circular dials on a control panel have a radius ratio of $2:3$ . If the circumference of the smaller dial is $12\pi \text{ centimeters}$ , what is the circumference of the larger dial?

8\pi \text{ cm}

18\pi \text{ cm}

36\pi \text{ cm}

81\pi \text{ cm}

Show answer & explanation

Answer: $18\pi \text{ cm}$

Answer

The circumference of the larger dial is

18\pi \text{ cm}

.

The ratio of the radii is 2:3. The smaller dial has a circumference of

12\pi

, meaning its radius is

6

(since

C = 2\pi r

). Setting up the proportion

\frac{6}{r} = \frac{2}{3}

gives a larger radius of

9

. The circumference of the larger dial is then

2\pi(9) = 18\pi \text{ cm}

.

Step-by-Step Solution

1

Determine the radius of the smaller dial.

2\pi r = 12\pi \rightarrow r = 6 \text{ cm}

The radius of the smaller dial is required to use the given ratio.

2

Apply the ratio to find the radius of the larger dial.

\frac{6}{r_{large}} = \frac{2}{3} \rightarrow 2r_{large} = 18 \rightarrow r_{large} = 9 \text{ cm}

The problem states the radius ratio between the smaller and larger dials is 2:3.

3

Calculate the circumference of the larger dial.

C = 2\pi(9) = 18\pi \text{ cm}

The question asks for the circumference of the larger dial.

Key Concept

Using proportions to find unknown circle dimensions and calculating circumference.

Alternative Method

Because circumference is directly proportional to radius (

C = 2\pi r

), the circumferences of two circles share the exact same ratio as their radii (

2:3

). Therefore, you can set up the proportion

\frac{12\pi}{C} = \frac{2}{3}

to solve directly for

C = 18\pi

without needing to calculate the specific radii.

Estimated Time:1m 0s

Question 130Question →

A right triangle has a hypotenuse that measures $20$ centimeters and a base that measures $12$ centimeters. What is the area of this triangle?

48\text{ cm}^2

120\text{ cm}^2

96\text{ cm}^2

192\text{ cm}^2

Show answer & explanation

Answer: $96\text{ cm}^2$

Answer

The area of the right triangle is

96\text{ cm}^2

.

To find the area of a right triangle, you need both the base and the height (the two legs). The Pythagorean theorem (

a^2 + b^2 = c^2

) reveals the missing height is

16

cm because

16^2 + 12^2 = 20^2

. Using the area formula (

A = \frac{1}{2}bh

), we multiply

\frac{1}{2} \times 12 \times 16

to get

96\text{ cm}^2

.

Step-by-Step Solution

1

Use the Pythagorean theorem (

a^2 + b^2 = c^2

) to find the missing height (the other leg).

a^2 + 12^2 = 20^2

The area formula requires the base and the height. We are given the hypotenuse (

20

) and the base (

12

).

2

Solve for the missing height (

a

).

a^2 + 144 = 400 \Rightarrow a^2 = 256 \Rightarrow a = 16

centimeters.

Subtracting

144

from

400

gives

256

, and the square root of

256

is

16

.

3

Calculate the area of the triangle using the formula

A = \frac{1}{2}bh

.

A = \frac{1}{2} \times 12 \times 16 = 96\text{ cm}^2

.

Substituting the base (

12

) and the newly found height (

16

) into the area formula gives the final answer.

Key Concept

Applying the Pythagorean theorem to find a missing side length, then calculating the triangle's area.

Estimated Time:1m 0s

Question 131Question →

A circle with center $O$ has points $A$ , $B$ , $C$ , and $D$ on its circumference.

• Line segment $AB$ passes through the center $O$ .
• Line segment $CD$ does not pass through $O$ .
• Line $L$ intersects the circle exactly at point $A$ .
• Line $M$ intersects the circle at points $C$ and $B$ , and extends infinitely in both directions.

Match each geometric term on the left with the notation that represents it on the right.

Click a left item, then click its matching right item

Show answer & explanation

Answer

Radius corresponds to Line segment OA; Diameter corresponds to Line segment AB; Chord corresponds to Line segment CD; Tangent corresponds to Line L; Secant corresponds to Line M.

Each geometric term strictly defines a specific type of line or segment related to a circle. By systematically checking the endpoints and intersection points described in the stem, we accurately pair each term to its corresponding notation.

Step-by-Step Solution

1

Evaluate Line segment OA

Matches Radius

Since O is the center and A is on the circle, the segment connecting them is the radius.

2

Evaluate Line segment AB

Matches Diameter

The segment connects two points on the circle and passes through the center O.

3

Evaluate Line segment CD

Matches Chord

The segment connects two points on the circle but does not pass through the center.

4

Evaluate Line L

Matches Tangent

The line touches the circle at exactly one point, which is the definition of a tangent line.

5

Evaluate Line M

Matches Secant

The line extends infinitely and intersects the circle at exactly two points.

Key Concept

Geometric definitions of circle components

Question 132Question →

A rectangular swimming pool has a diagonal length of $20\text{ meters}$ and a width of $12\text{ meters}$ . What is the perimeter of the swimming pool?

16\text{ meters}

48\text{ meters}

56\text{ meters}

64\text{ meters}

Show answer & explanation

Answer: $56\text{ meters}$

Answer

56\text{ meters}

The diagonal of a rectangle creates a right triangle where the diagonal is the hypotenuse. Using the Pythagorean theorem (

a^2 + b^2 = c^2

), the missing length is

\sqrt{20^2 - 12^2} = \sqrt{400 - 144} = \sqrt{256} = 16\text{ meters}

. The perimeter of the rectangle is the total distance around the outside, which is calculated as

2 \times (\text{width} + \text{length}) = 2 \times (12 + 16) = 2 \times 28 = 56\text{ meters}

.

Step-by-Step Solution

1

Identify the geometric relationship between the dimensions.

The width, length, and diagonal of a rectangle form a right triangle where the diagonal is the hypotenuse.

Rectangles have right angles at all corners, allowing the use of the Pythagorean theorem.

2

Use the Pythagorean theorem to find the length of the pool.

12^2 + \text{length}^2 = 20^2

, which becomes

144 + \text{length}^2 = 400

. Subtracting

144

gives

\text{length}^2 = 256

, so the length is

16\text{ meters}

.

The Pythagorean theorem (

a^2 + b^2 = c^2

) allows us to find a missing side of a right triangle when two sides are known.

3

Calculate the perimeter of the rectangle.

Perimeter =

2 \times (\text{width} + \text{length}) = 2 \times (12 + 16) = 2 \times 28 = 56\text{ meters}

.

The question asks for the perimeter of the entire rectangular pool, which requires combining all four outer sides.

Key Concept

Applying the Pythagorean theorem to find missing dimensions before calculating perimeter.

Estimated Time:1m 0s

Question 133Question →

Provide the missing numerical values to complete the calculations for the stage construction.

Fill in the blanks below

A theater crew is reinforcing a rectangular stage that measures

16

feet in width and

30

feet in length. They place a diagonal steel support beam underneath, running directly from one corner to the opposite corner. By applying the Pythagorean theorem, the length of this diagonal support beam is calculated to be feet. Consequently, the total perimeter of the right triangle formed by the width, length, and the new diagonal beam is feet.

Show answer & explanation

Answer

The diagonal support beam is 34 feet long, and the total perimeter of the right triangle is 80 feet.

The width and length of the rectangular stage act as the two legs of a right triangle. Using the Pythagorean theorem (

a^2 + b^2 = c^2

), we substitute the known legs:

16^2 + 30^2 = 256 + 900 = 1156

. The square root of

1156

is

34

, so the diagonal hypotenuse is

34

feet. The perimeter of this triangle is the sum of its three sides:

16 + 30 + 34 = 80

feet.

Step-by-Step Solution

1

Identify the lengths of the two legs of the right triangle formed by the stage's dimensions.

Leg a = 16 feet, Leg b = 30 feet.

The width and length of a rectangle meet at a 90-degree angle, serving as the legs of the right triangle.

2

Apply the Pythagorean theorem (

a^2 + b^2 = c^2

) to calculate the hypotenuse.

16^2 + 30^2 = 256 + 900 = 1156

. Taking the square root,

c = \sqrt{1156} = 34

feet.

The diagonal beam forms the hypotenuse of the right triangle spanning opposite corners.

3

Calculate the perimeter of the newly formed right triangle.

Perimeter =

16 + 30 + 34 = 80

feet.

The perimeter of a triangle is the total sum of its three side lengths.

Key Concept

Applying the Pythagorean theorem to find a missing hypotenuse, and calculating the perimeter of a right triangle.

Question 134Question →

A circular steel plate is cut into $6$ equal sectors. If the total area of the original plate was $144\pi \text{ cm}^2$ , what is the arc length of the outer edge of one sector?

4\pi \text{ cm}

2\pi \text{ cm}

24\pi \text{ cm}

12\pi \text{ cm}

Show answer & explanation

Answer: $4\pi \text{ cm}$

Answer

The arc length of one sector is

4\pi \text{ cm}

.

The correct answer is found by first using the total area (

144\pi

) to find the radius (

r = 12

). Next, the full circumference is calculated using

C = 2\pi r

, resulting in

24\pi

. Finally, because the plate is divided into 6 equal sectors, the full circumference is divided by 6, giving an arc length of

4\pi \text{ cm}

.

Step-by-Step Solution

1

Find the radius of the original circular plate.

Using the area formula

A = \pi r^2

, substitute the given area:

144\pi = \pi r^2

. Dividing by

\pi

gives

r^2 = 144

, so the radius

r = 12 \text{ cm}

.

The radius is needed to calculate the total circumference of the circle.

2

Calculate the full circumference of the plate.

Using the circumference formula

C = 2\pi r

, substitute

r = 12

to get

C = 2\pi(12) = 24\pi \text{ cm}

.

The arc length of a sector is a fraction of the circle's total circumference.

3

Determine the arc length of one of the 6 equal sectors.

Divide the full circumference by 6:

\frac{24\pi}{6} = 4\pi \text{ cm}

.

Since the plate is cut into 6 equal sectors, the outer edge of one sector is exactly one-sixth of the total circumference.

Key Concept

The relationship between a circle's area, its circumference, and the proportional arc length of its sectors.

Question 135Question →

A custom corner desk is shaped like a right triangle so it fits perfectly against two perpendicular walls. The sections of the desk that touch the walls measure $24\text{ inches}$ and $10\text{ inches}$ in length. What is the length of the front edge of the desk in inches?

Show answer & explanation

Answer: 26

Answer

26

Because the walls meet at a perpendicular right angle, the corner desk forms a right triangle. The edges touching the walls act as the legs (

a = 24

,

b = 10

), and the front edge is the hypotenuse (

c

). Using the Pythagorean Theorem (

a^2 + b^2 = c^2

), we square both legs to get

576

and

100

. Adding these together gives

676

. The square root of

676

is

26

, which is the exact length of the front edge in inches.

Step-by-Step Solution

1

Identify the known sides of the right triangle

Legs are

a = 24

and

b = 10

.

The walls are perpendicular, forming a

90^{\circ}

angle. Therefore, the edges touching the walls are the legs of a right triangle.

2

Set up the Pythagorean Theorem

24^2 + 10^2 = c^2

The Pythagorean Theorem (

a^2 + b^2 = c^2

) is the standard formula used to find the missing hypotenuse (

c

) when both legs are known.

3

Square the lengths of the legs

576 + 100 = c^2

Evaluating the exponents is the required next step in the order of operations.

4

Add the squared values

676 = c^2

Combining the squares of the legs gives the square of the hypotenuse.

5

Find the square root of the sum

c = 26

Taking the square root of

676

isolates

c

, revealing the true length of the desk's front edge.

Key Concept

Using the Pythagorean Theorem (

a^2 + b^2 = c^2

) to calculate the length of a hypotenuse.

Alternative Method

You can solve this rapidly without large calculations by recognizing special Pythagorean triples. This triangle is a scaled-up version of the fundamental

5-12-13

right triangle. Multiplying each side of a

5-12-13

triangle by

2

gives side lengths of

10

,

24

, and

26

. Since the legs are

10

and

24

, the hypotenuse must be

26

.

Estimated Time:45s

Question 136Question →

The circular foundation of a radar dome has a circumference of $36\pi \text{ feet}$ . What is the area of this circular foundation, in square feet?

(Enter only the numerical coefficient of $\pi$ in your answer. For example, if the area is $100\pi$ , enter 100.)

Show answer & explanation

Answer: 324

Answer

324

The circumference of a circle is calculated using

C = 2\pi r

. Setting the given circumference

36\pi = 2\pi r

gives a radius of

18 \text{ feet}

. The area is found using

A = \pi r^2

, which becomes

A = \pi(18)^2 = 324\pi \text{ square feet}

. Since the question asks for the numerical coefficient of

\pi

, the correct value to enter is 324.

Step-by-Step Solution

1

Find the radius of the circular foundation from its given circumference.

r = 18

The circumference formula is

C = 2\pi r

. Setting

2\pi r = 36\pi

allows us to solve for the radius.

2

Calculate the area of the foundation using the radius.

A = 324\pi

The area formula is

A = \pi r^2

. Substituting

r = 18

gives

A = \pi(18)^2 = 324\pi

.

3

Extract the numerical coefficient of

\pi

for the final answer.

324

The question explicitly asks to enter only the numerical coefficient of

\pi

.

Key Concept

Calculating the area of a circle by first deriving the radius from its circumference

Question 137Question →

A solid rectangular block has a square base with an area of $25$ square inches. If the total volume of the block is $150$ cubic inches, what is the total surface area of the block in square inches?

150

200

170

220

Show answer & explanation

Answer: $170$

Answer

170

To find the total surface area, all dimensions of the block must be known. The side length of the square base is

\sqrt{25} = 5

inches. The height is the volume divided by the base area:

150 \div 25 = 6

inches. The total surface area is the sum of the areas of all six faces: two square bases (

2 \times 25 = 50

) and four identical rectangular sides (

4 \times 5 \times 6 = 120

). The total is

50 + 120 = 170

square inches.

Step-by-Step Solution

1

Find the side length of the square base.

5

inches

The area of a square is the side length squared (

s^2 = 25

), so the side length is the square root of the area.

2

Find the height of the rectangular block.

6

inches

The volume of a rectangular block is the base area multiplied by the height (

V = B \times h

). Dividing the volume (

150

) by the base area (

25

) gives the height.

3

Calculate the total surface area of the block.

170

square inches

The total surface area is the sum of the two square bases (

2 \times 25 = 50

) and the four identical rectangular sides (

4 \times 5 \times 6 = 120

). Adding these together yields

50 + 120 = 170

.

Key Concept

Volume and Surface Area of Rectangular Prisms

Question 138Question →

A right-angled triangular bracket is used to support a shelf. The longest side (hypotenuse) of the bracket has a length represented by the expression $2x + 1$ inches, and the horizontal leg has a length represented by $3x - 6$ inches. If $x = 6$ , what is the length of the vertical leg?

1

5

25

313

Show answer & explanation

Answer: 5

Answer

The vertical leg has a length of

5

inches.

After substituting

x = 6

, the hypotenuse is

13

inches and the horizontal leg is

12

inches. Using the Pythagorean theorem (

a^2 + b^2 = c^2

), we get

b^2 + 12^2 = 13^2

. Solving for

b

yields

b^2 = 25

, which means

b = 5

.

Step-by-Step Solution

1

Substitute

x = 6

into the expression for the hypotenuse.

2(6) + 1 = 12 + 1 = 13\text{ inches}

To find the numerical length of the longest side.

2

Substitute

x = 6

into the expression for the horizontal leg.

3(6) - 6 = 18 - 6 = 12\text{ inches}

To find the numerical length of the given leg.

3

Apply the Pythagorean theorem to find the vertical leg (let's call it

b

).

b^2 + 12^2 = 13^2

The Pythagorean theorem relates the three sides of a right triangle:

a^2 + b^2 = c^2

.

4

Solve for

b

.

b^2 + 144 = 169 \Rightarrow b^2 = 169 - 144 = 25 \Rightarrow b = 5\text{ inches}

To calculate the final missing side length.

Key Concept

To find a missing leg in a right triangle when given the hypotenuse and the other leg, substitute the known values into the Pythagorean theorem (

a^2 + b^2 = c^2

) and solve for the unknown leg by subtracting the squares and taking the square root.

Alternative Method

Recognize that

5

,

12

, and

13

form a common Pythagorean triple. Once the hypotenuse (

13

) and one leg (

12

) are calculated, the other leg must immediately be

5

.

Estimated Time:1m 15s

Question 139Question →

Pair the given geometric descriptions of 3D figures with their corresponding exact volumes.

Click a left item, then click its matching right item

Show answer & explanation

Answer

The cylinder matches

80\pi

, the rectangular prism matches

80

, the cube matches

64

, and the sphere matches

36\pi

.

Matching each figure to its correct volume requires applying the specific volume formula for that geometric shape: cylinder (

V = \pi r^2h

), rectangular prism (

V = lwh

), cube (

V = s^3

), and sphere (

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

).

Step-by-Step Solution

1

Calculate the volume of the cylinder.

V = \pi(4^2)(5) = 80\pi

Applying the cylinder volume formula

V = \pi r^2h

.

2

Calculate the volume of the rectangular prism.

V = 8 \times 5 \times 2 = 80

Applying the rectangular prism volume formula

V = lwh

.

3

Calculate the volume of the cube.

V = 4^3 = 64

Applying the cube volume formula

V = s^3

.

4

Calculate the volume of the sphere.

V = \frac{4}{3}\pi(3^3) = 36\pi

Applying the sphere volume formula

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

.

Key Concept

Volume and Surface Area of 3D Figures

Question 140Question →

Read the scenario and provide the missing numerical value to complete the sentence.

Fill in the blanks below

A guy wire is securely attached to a vertical radio mast at a height of

60\text{ feet}

above the level ground. The other end of the wire is anchored to the ground exactly

25\text{ feet}

from the base of the mast. Assuming the mast is perfectly perpendicular to the ground, the length of the guy wire is feet.

Show answer & explanation

Answer

65

The mast, the ground, and the guy wire form a right triangle where the mast and the ground act as the two legs, and the guy wire is the hypotenuse. Using the Pythagorean theorem (

a^2 + b^2 = c^2

), we substitute the given leg lengths:

25^2 + 60^2 = c^2

. This simplifies to

625 + 3600 = 4225

. The square root of

4225

is

65

, meaning the guy wire is

65\text{ feet}

long. Additionally, recognizing that the side lengths (

25, 60, 65

) are a multiple of the common

5-12-13

Pythagorean triple (each multiplied by

5

) provides a quick way to solve the problem without large calculations.

Step-by-Step Solution

1

Identify the shape formed by the scenario.

A right triangle is formed by the vertical mast, the level ground, and the diagonal guy wire.

The mast is perpendicular to the ground, creating a 90-degree angle.

2

Set up the Pythagorean theorem equation.

a^2 + b^2 = c^2

, where

a = 25

and

b = 60

.

The legs of the right triangle are the distance along the ground (25 ft) and the height up the mast (60 ft). We need to find the hypotenuse,

c

.

3

Calculate the squares of the leg lengths.

25^2 = 625

and

60^2 = 3600

.

Squaring the values is required before adding them to find the square of the hypotenuse.

4

Add the squared values together.

625 + 3600 = 4225

.

This sum represents

c^2

, the square of the guy wire's length.

5

Find the square root of the sum to determine the final length.

\sqrt{4225} = 65

.

Taking the square root of

c^2

gives the actual length of the hypotenuse,

c

.

Key Concept

Pythagorean Theorem Application

Geometry

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