Applied Geometry Practice Questions — Armed Services Vocational Aptitude Battery (ASVAB) — Page 4

Question 61Question →

A large cylindrical water cistern used for irrigation has a diameter of $10\text{ feet}$ and a depth of $6\text{ feet}$ . A heavy-duty pump removes water from the completely full cistern at a constant rate of $31.4\text{ cubic feet}$ per minute. Using $3.14$ for $\pi$ , how many minutes will it take to empty the cistern?

3

6

15

60

Show answer & explanation

Answer: 15

Answer

15

To find the time it takes to empty the cistern, we first need its total volume. The radius is half the diameter, so

r = 10 / 2 = 5\text{ feet}

. The formula for the volume of a cylinder is

V = \pi \times r^2 \times h

. Plugging in our values:

V = 3.14 \times 5^2 \times 6 = 3.14 \times 25 \times 6 = 3.14 \times 150 = 471\text{ cubic feet}

. The pump removes

31.4\text{ cubic feet}

of water each minute. Dividing the total volume by the rate gives the time:

471 / 31.4 = 15\text{ minutes}

.

Step-by-Step Solution

1

Find the radius of the cistern.

r = 10 / 2 = 5\text{ feet}

The volume formula requires the radius, which is half of the given diameter.

2

Calculate the total volume of the cistern using

V = \pi \times r^2 \times h

.

V = 3.14 \times 5^2 \times 6 = 3.14 \times 25 \times 6 = 3.14 \times 150 = 471\text{ cubic feet}

This determines the total amount of water that needs to be pumped out.

3

Determine the time required to empty the tank by dividing the total volume by the pump rate.

Time = 471 / 31.4 = 15\text{ minutes}

Dividing the total amount of water by the rate of removal gives the total time.

Key Concept

Cylindrical Volume Applications

Question 62Question →

A hotel maintenance team is installing a new cylindrical hot water heater. The interior of the heater has a diameter of $4\text{ feet}$ and a height of $7\text{ feet}$ . Using $\frac{22}{7}$ for $\pi$ , how many minutes will it take to completely fill the empty heater if water flows in at a constant rate of $4\text{ cubic feet}$ per minute?

Show answer & explanation

Answer: 22

Answer

It will take 22 minutes to completely fill the hot water heater.

The correct answer is found by first calculating the volume of the cylindrical heater using the formula

V = \pi r^2 h

. Given the diameter of

4\text{ feet}

, the radius is

2\text{ feet}

. Substituting the given values:

V = \frac{22}{7} \times 2^2 \times 7 = 88\text{ cubic feet}

. Dividing this total volume by the fill rate of

4\text{ cubic feet}

per minute gives the total time:

88 \div 4 = 22\text{ minutes}

.

Step-by-Step Solution

1

Determine the radius of the cylindrical heater.

Radius =

2\text{ feet}

The volume formula requires the radius, which is half of the given diameter (

4\text{ feet}

).

2

Calculate the total volume of the cylinder.

Volume =

88\text{ cubic feet}

Using the formula

V = \pi r^2 h

with

r = 2

,

h = 7

, and

\pi = \frac{22}{7}

, we get

V = \frac{22}{7} \times 2^2 \times 7 = 88

.

3

Calculate the time required to fill the heater.

Time =

22\text{ minutes}

Dividing the total volume (

88\text{ cubic feet}

) by the flow rate (

4\text{ cubic feet}

per minute) gives the required time.

Key Concept

Cylindrical Volume Applications

Question 63Question →

A telecommunications crew is installing a straight fiber-optic cable from the base of a network tower to a control panel mounted on a nearby building. The horizontal distance from the tower base to the building is $12$ yards. The control panel is mounted $5$ yards above the ground. How many feet long must the cable be to reach exactly from the tower base to the control panel?

13

17

39

51

Show answer & explanation

Answer: 39

Answer

39

The scenario creates a right triangle with legs measuring

5

yards and

12

yards. Using the Pythagorean theorem (

a^2 + b^2 = c^2

), we find the hypotenuse is

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

yards. Because the question asks for the answer in feet, we multiply

13

yards by

3

feet per yard to get exactly

39

feet.

Step-by-Step Solution

1

Identify the shape formed by the scenario.

The ground, the building wall, and the cable form a right triangle.

This allows the use of the Pythagorean theorem (

a^2 + b^2 = c^2

) to find the length of the diagonal cable.

2

Calculate the length of the cable in yards.

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

. The square root of

169

is

13

yards.

Applying the Pythagorean theorem using the given leg lengths of

5

yards and

12

yards yields the hypotenuse.

3

Convert the final length from yards to feet.

13 \text{ yards} \times 3 \text{ feet/yard} = 39 \text{ feet}

.

The question specifically asks for the length of the cable in feet.

Key Concept

Pythagorean Theorem and Unit Conversions

Estimated Time:1m 0s

Question 64Question →

A military training field is rectangular, measuring $160$ yards wide and $300$ yards long. A recruit must travel from one corner of the field to the opposite corner. If the recruit runs directly across the diagonal instead of moving along the outer edges of the two sides, how many yards does the recruit save?

Show answer & explanation

Answer: 120

Answer

120 yards

The route along the edges is the sum of the width and length (

160 + 300 = 460

yards). The diagonal route is the hypotenuse of the right triangle formed by these sides, calculated as

\sqrt{160^2 + 300^2} = 340

yards. The distance saved is

460 - 340 = 120

yards.

Step-by-Step Solution

1

Calculate the total distance if traveling along the two outer edges of the rectangular field.

160 + 300 = 460

yards.

This establishes the longer route distance to compare against the diagonal shortcut.

2

Set up the Pythagorean theorem to find the length of the diagonal route.

160^2 + 300^2 = c^2

The rectangular field forms a right triangle where the width and length are the legs, and the diagonal is the hypotenuse.

3

Solve for the diagonal length (

c

).

c = \sqrt{115600} = 340

yards.

This gives the exact distance of the direct route.

4

Calculate the difference between the two routes.

460 - 340 = 120

yards.

Subtracting the shorter diagonal route from the longer perimeter route yields the exact distance saved.

Key Concept

Applying the Pythagorean theorem (

a^2 + b^2 = c^2

) to find a direct diagonal distance, and comparing it to the linear distance along the edges.

Question 65Question →

A surveyor is mapping a plot of land and launches a drone from a perfectly flat field. The drone ascends vertically to an altitude of $30$ meters. It then travels horizontally in a straight line for $16$ meters to photograph a landmark. What is the direct line-of-sight distance, in meters, from the launch point on the ground to the drone?

Show answer & explanation

Answer: 34

Answer

The direct line-of-sight distance is

34

meters.

The drone's flight path forms a right-angled triangle relative to the starting point. By applying the Pythagorean theorem (

a^2 + b^2 = c^2

), we square the vertical distance (

30^2 = 900

) and the horizontal distance (

16^2 = 256

). Adding these values yields

1156

. Taking the square root of

1156

gives the direct line-of-sight distance, which is

34

meters.

Step-by-Step Solution

1

Model the drone's flight path as a right-angled triangle.

The vertical ascent forms one leg (

a = 30

) and the horizontal flight forms the other leg (

b = 16

). The direct line-of-sight distance is the hypotenuse (

c

).

Vertical and horizontal movements are perpendicular, creating a

90^\circ

angle.

2

Apply the Pythagorean theorem to calculate the hypotenuse.

a^2 + b^2 = c^2

becomes

30^2 + 16^2 = c^2

.

The Pythagorean theorem relates the lengths of the legs to the length of the hypotenuse in any right triangle.

3

Square the lengths of the two legs and add them together.

900 + 256 = 1156

. So,

c^2 = 1156

.

This evaluates the left side of the equation to isolate

c^2

.

4

Take the square root of the sum to find the actual distance.

c = \sqrt{1156} = 34

.

Taking the square root of

c^2

yields the length of the hypotenuse.

Key Concept

Applying the Pythagorean theorem to find the hypotenuse in a practical right-triangle scenario.

Question 66Question →

A commercial facility is securing a right-triangular storage lot. The two perpendicular boundaries of the lot measure $60$ yards and $80$ yards. Security fencing will be installed along the entire perimeter to create a closed loop. If a support post must be placed every $12$ feet, how many total posts are required to complete the fence?

20

35

60

200

Show answer & explanation

Answer: 60

Answer

60

The correct answer is found by first determining the hypotenuse of the right-triangular lot, which is 100 yards (

60^2 + 80^2 = 100^2

). The total perimeter is 240 yards (60 + 80 + 100). Because the post spacing is given in feet, the perimeter must be converted to 720 feet (240 × 3). Finally, dividing the 720-foot boundary by the 12-foot spacing requirement yields exactly 60 posts.

Step-by-Step Solution

1

Find the length of the third side of the lot using the Pythagorean theorem (

a^2 + b^2 = c^2

).

60^2 + 80^2 = 3600 + 6400 = 10000

. The square root of

10000

is

100

yards.

The boundary of a right-triangular lot includes the hypotenuse, which must be found to calculate the total perimeter.

2

Calculate the total perimeter of the triangular lot.

60 + 80 + 100 = 240

yards.

The fencing will enclose the entire boundary of the lot.

3

Convert the perimeter into feet to match the unit of the post spacing.

240 \text{ yards} \times 3 \text{ feet/yard} = 720 \text{ feet}

.

Measurements must be in the same unit (feet) before dividing.

4

Divide the total perimeter by the distance required between each post.

720 \div 12 = 60

posts.

For a closed loop, the total number of posts equals the total perimeter divided by the spacing distance.

Key Concept

Calculating the perimeter of a right triangle using the Pythagorean theorem, converting units of measurement, and determining spacing requirements for closed boundaries.

Alternative Method

Instead of converting the perimeter to feet, convert the post spacing to yards. Since 3 feet equals 1 yard, 12 feet equals 4 yards. Divide the 240-yard perimeter by 4 yards to get 60 posts.

Estimated Time:1m 30s

Question 67Question →

A renewable energy company is installing a chain-link fence around a newly constructed rectangular solar farm. The solar farm measures $120$ yards in length and $85$ yards in width. The fence will enclose the entire perimeter, with the exception of two access gates that each measure $5$ yards wide. If the fencing material costs $\$22$ per linear yard, what is the total cost of the fencing material in dollars?

Show answer & explanation

Answer: 8800

Answer

The total cost of the fencing material is 8800 dollars.

The total perimeter of the solar farm is

410

yards (

120 + 120 + 85 + 85

). Subtracting the

10

yards for the two

5

-yard gates leaves

400

yards of fencing to be installed. Multiplying

400

yards by the

\$22

per yard cost gives a total of

\$8800

.

Step-by-Step Solution

1

Calculate the total perimeter of the rectangular solar farm.

P = 2 \times (120 + 85) = 2 \times 205 = 410

yards.

The perimeter formula for a rectangle is

P = 2(L + W)

, which represents the total boundary length before any openings are considered.

2

Determine the total width of the two access gates.

2 \times 5 = 10

yards.

Since there are two gates and each is

5

yards wide, their combined width must be subtracted from the total perimeter.

3

Calculate the actual length of fencing needed.

410 - 10 = 400

yards.

The fencing covers the perimeter minus the space left open for the access gates.

4

Calculate the total cost of the fencing material.

400 \times 22 = 8800

.

Multiplying the total yards of fencing by the cost per yard gives the total cost in dollars.

Key Concept

Calculating the perimeter of a rectangle, subtracting openings, and finding the total cost based on a unit price.

Question 68Question →

A homeowner is planning to enclose a rectangular residential lot with a new privacy fence. The lot has a length of $20$ yards and a width of $45$ feet. The fencing material costs $\$12.50$ per linear foot. If the homeowner needs to leave a $6$ -foot opening for a driveway gate, what will be the total cost of the fencing materials?

\$255

\$1,550

\$2,550

\$33,675

Show answer & explanation

Answer: $\$2,550$

Answer

The total cost of the fencing materials is

\$2,550

.

The correct answer is found by first converting all units to feet. The length of

20

yards is equal to

60

feet. The total perimeter of the rectangular lot is

2 \times (60 + 45) = 210

feet. Subtracting the

6

-foot gap for the gate leaves

204

linear feet of fencing required. Multiplying this by the cost of

\$12.50

per foot yields a total cost of

\$2,550

.

Step-by-Step Solution

1

Convert the length of the lot from yards to feet.

20 \text{ yards} \times 3 = 60 \text{ feet}

All measurements must be in the same unit (feet) because the fencing cost is given per linear foot.

2

Calculate the total perimeter of the rectangular lot.

P = 2 \times (60 + 45) = 2 \times 105 = 210 \text{ feet}

The fencing will go around the entire boundary of the rectangular lot.

3

Subtract the length of the driveway gate from the total perimeter.

210 - 6 = 204 \text{ feet}

Fencing material is not needed for the space where the gate will be installed.

4

Calculate the total cost by multiplying the fencing length by the cost per foot.

204 \times \$12.50 = \$2,550

The total cost is the required length in feet multiplied by the price per linear foot.

Key Concept

Calculating the perimeter of a rectangle with mixed units and adjusting for an opening.

Question 69Question →

A military engineering unit is installing a security fence around a rectangular vehicle depot. The depot is $150$ yards long and $80$ yards wide. The fencing plan includes leaving a single $30$ -foot wide opening for a main access gate. If the fencing material costs $\$15$ per linear foot, what is the total cost of the fencing material needed? (Do not include commas or dollar signs in your answer)

Show answer & explanation

Answer: 20250

Answer

20250

To find the total cost, first determine the total perimeter of the depot in yards:

2 \times (150 + 80) = 460

yards. Since the fencing cost and gate width are in feet, convert the perimeter to feet:

460 \times 3 = 1380

feet. Next, subtract the

30

-foot gate opening where no fencing is needed:

1380 - 30 = 1350

feet of fencing. Finally, multiply the required fencing by the cost per foot:

1350 \times \$15 = \$20,250

.

Step-by-Step Solution

1

Calculate the perimeter of the rectangular depot in yards.

2(150 + 80) = 2(230) = 460

yards.

The fencing will enclose the boundary of the rectangular area.

2

Convert the perimeter from yards to feet.

460 \times 3 = 1380

feet.

The cost of the fencing is given per linear foot and the gate is measured in feet, so all measurements must be in feet.

3

Subtract the gate opening from the total perimeter.

1380 - 30 = 1350

feet.

Fencing material is not needed for the gate opening.

4

Calculate the total cost of the fencing material.

1350 \times 15 = 20250

dollars.

The total length of fencing required is multiplied by the cost per linear foot.

Key Concept

Calculating perimeter, converting units (yards to feet), and determining total cost based on a linear rate.

Question 70Question →

An event coordinator is setting up a temporary security perimeter for an outdoor festival. The secured VIP zone is rectangular, measuring $30$ yards in length and $20$ yards in width. An un-fenced opening of $5$ yards must be left on one side for the main entrance gate. If the temporary fencing costs $\$4.50$ per linear foot, what is the total cost, in dollars, of the fencing required?

Show answer & explanation

Answer: 1282.5

Answer

1282.5

The correct total cost is

1282.50. To find this, the full perimeter of the 30-by-20 yard rectangle is calculated as 100 yards. After subtracting the 5-yard opening, 95 yards of fencing are needed. Because the fencing is priced per foot, the 95 yards must be converted to 285 feet by multiplying by 3. Finally, multiplying 285 feet by the

4.50 per foot cost yields 1282.50.

Step-by-Step Solution

1

Calculate the total perimeter of the rectangular zone in yards.

100 yards

The perimeter represents the total distance around the rectangle before accounting for any openings.

2

Subtract the gate opening from the perimeter.

95 yards

Fencing is not required for the 5-yard entrance gate.

3

Convert the required fencing length from yards to feet.

285 feet

The cost of the fencing is provided per linear foot, so the length must be in feet.

4

Calculate the total cost by multiplying the length in feet by the price per foot.

1282.5

Total cost equals the total amount of material multiplied by the unit price.

Key Concept

Calculating perimeter of a rectangle, adjusting for openings, and converting units for cost calculation.

Question 71Question →

A logistics company is securing an L-shaped equipment depot with a chain-link fence. The facility has a 120-foot southern border, a 90-foot western border, a 75-foot northern border, and a 45-foot eastern border. The remaining two inner boundary walls connect the northern and eastern borders to complete the L-shape. The entire perimeter will be fenced except for a 15-foot wide entrance gate on the southern border. If the fencing material costs $25$ per yard, what is the total cost of the fencing?

$2,625

$3,375

$10,125

$24,375

Show answer & explanation

Answer: $3,375

Answer

The total cost of the fencing is $3,375.

The correct answer is found by first determining the missing inner dimensions of the L-shape (120 - 75 = 45 feet, and 90 - 45 = 45 feet) to find the total 6-sided perimeter (420 feet). After subtracting the 15-foot gate, 405 feet of fencing is needed. Converting this to yards by dividing by 3 gives 135 yards. Multiplying 135 yards by the

25 per yard cost results in

3,375.

Step-by-Step Solution

1

Calculate the lengths of the two missing inner boundaries.

Inner horizontal border = 120 - 75 = 45 feet. Inner vertical border = 90 - 45 = 45 feet.

All six sides of the L-shape are needed to find the total perimeter. The missing sides are found by subtracting the known parallel outer edges.

2

Calculate the total perimeter of the equipment depot.

120 + 90 + 75 + 45 + 45 + 45 = 420 feet.

The perimeter is the total length of the continuous boundary line enclosing the facility.

3

Subtract the width of the entrance gate from the total perimeter.

420 - 15 = 405 feet.

The 15-foot gate opening does not require fencing material, so it must be removed from the total.

4

Convert the required fencing from feet to yards.

405 / 3 = 135 yards.

The fencing cost is provided in yards, so the perimeter measurement must match the pricing unit.

5

Calculate the total cost of the fencing material.

135 \times 25 = 3,375 dollars.

Multiplying the total yards of fencing by the cost per yard gives the final price.

Key Concept

Calculating the perimeter of composite rectilinear shapes, subtracting non-fenced gaps, and applying unit conversions for total cost.

Question 72Question →

A contractor is installing a security perimeter around a rectangular warehouse facility. The facility is $40$ yards long and $25$ yards wide. The fence will include three $10$ -foot gates that do not require fencing material. If the fencing costs $\$18$ per yard, what is the total cost of the fencing material required?

$990

$1,800

$2,160

$18,000

Show answer & explanation

Answer: $2,160

Answer

The correct total cost of the fencing material is $2,160.

The total perimeter is found using the formula

2 \times (L + W)

, yielding

130

yards. The three

10

-foot gates have a combined width of

30

feet, which converts to

10

yards (

30 \div 3

). Subtracting the gate width leaves

120

yards of actual fencing needed. Multiplying this by the cost of

\$18

per yard yields the total of

\$2,160

.

Step-by-Step Solution

1

Calculate the total perimeter of the rectangular facility in yards.

2 \times (40 + 25) = 130

yards

The fencing will go around the entire boundary of the rectangle, which requires the perimeter formula

P = 2L + 2W

.

2

Convert the total width of the gates from feet to yards.

3 \text{ gates} \times 10 \text{ feet/gate} = 30 \text{ feet}

. Since

3 \text{ feet} = 1 \text{ yard}

,

30 \text{ feet} = 10 \text{ yards}

.

The perimeter and the fencing cost are both in yards, so the gate measurements must match the primary unit.

3

Subtract the gate width from the total perimeter to find the required fencing.

130 \text{ yards} - 10 \text{ yards} = 120 \text{ yards}

The sections where the gates are located do not require any fencing material.

4

Calculate the total cost of the fencing material.

120 \text{ yards} \times \$18/\text{yard} = \$2,160

Multiplying the total required yards by the price per yard provides the final monetary cost.

Key Concept

Calculating perimeter of a rectangle, performing unit conversions (feet to yards), and subtracting openings to find total material cost.

Question 73Question →

An airport authority is installing a security barrier around a new rectangular radar facility. The facility measures $40$ yards in length and $32$ yards in width. The design for the security barrier requires a heavy-duty steel post at each of the four corners, with additional posts placed exactly every $8$ feet along all four sides. How many steel posts are required in total to enclose the facility?

Show answer & explanation

Answer: 54

Answer

54 posts are required in total.

The correct answer is found by first ensuring all units match. Converting the dimensions to feet gives a length of 120 feet and a width of 96 feet. The total perimeter is 432 feet. Dividing this continuous perimeter by the 8-foot spacing requirement yields exactly 54 posts.

Step-by-Step Solution

1

Convert the given dimensions from yards to feet.

Length = 40 * 3 = 120 feet. Width = 32 * 3 = 96 feet.

The post spacing is provided in feet, so all measurements must be converted to the same unit before calculating the perimeter.

2

Calculate the total perimeter of the rectangular facility.

Perimeter = 2 * (120 + 96) = 2 * 216 = 432 feet.

The security barrier encloses the entire outer boundary of the facility, which corresponds to its perimeter.

3

Determine the total number of posts required.

Total posts = 432 / 8 = 54.

For any closed loop, the total number of equally spaced posts equals the total perimeter divided by the distance between the posts.

Key Concept

Calculating perimeter of a rectangle, unit conversions, and dividing a perimeter into equal segments.

Question 74Question →

A rectangular training pool at a military base is $40$ feet long and $25$ feet wide. It is initially filled with water to a uniform depth of $8$ feet. If $1,500$ cubic feet of water is pumped out of the pool, what is the new depth of the water, in feet?

Show answer & explanation

Answer: 6.5

Answer

6.5

The pool's base has an area of

40 \times 25 = 1,000

square feet. When

1,500

cubic feet of water is removed, the water level drops by

1,500 \div 1,000 = 1.5

feet. Subtracting this drop from the initial depth of

8

feet gives a new depth of

6.5

feet. Alternatively, finding the original volume (

8,000

cubic feet), subtracting the removed water (

6,500

cubic feet), and dividing by the area (

1,000

square feet) also yields

6.5

feet.

Step-by-Step Solution

1

Calculate the base area of the rectangular pool.

1,000

square feet

The base area is needed to determine how changes in volume affect the depth of the water. Area = length

\times

width =

40 \times 25 = 1,000

.

2

Determine the change in the water depth.

The depth drops by

1.5

feet.

Dividing the removed volume by the base area gives the vertical drop in water level.

1,500 \div 1,000 = 1.5

.

3

Calculate the new depth.

6.5

feet

Subtract the drop in water level from the initial depth.

8 - 1.5 = 6.5

.

Key Concept

Calculating changes in the depth of a rectangular volume when a specific amount of capacity is removed.

Question 75Question →

A zip line is being installed from an observation deck down to a ground anchor. The observation deck is $45$ yards directly above the base of the structure. The ground anchor is secured on flat ground $24$ yards horizontally away from the base. Assuming the zip line is pulled perfectly straight, what is the required length of the zip line cable in feet?

51

69

153

207

Show answer & explanation

Answer: 153

Answer

153

The correct answer accurately calculates the diagonal length using the Pythagorean theorem (

45^2 + 24^2 = c^2

, so

c = 51

yards) and then converts that distance into feet by multiplying by 3 (

51 \times 3 = 153

feet).

Step-by-Step Solution

1

Apply the Pythagorean theorem (

a^2 + b^2 = c^2

) to find the diagonal length of the zip line in yards.

24^2 + 45^2 = 576 + 2025 = 2601

square yards.

The vertical height and horizontal distance form a right triangle with the zip line acting as the hypotenuse.

2

Calculate the square root of the sum to find the actual length.

\sqrt{2601} = 51

yards.

This isolates

c

(the hypotenuse) to give the actual length of the zip line in yards.

3

Convert the length from yards to feet.

51 \times 3 = 153

feet.

The question specifically asks for the length in feet, and there are

3

feet in one yard.

Key Concept

Using the Pythagorean theorem to find a hypotenuse and converting units of measurement.

Applied Geometry

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