Cylindrical Volume Applications Alıştırma Soruları — Armed Services Vocational Aptitude Battery (ASVAB)

Soru 1Soru →

A cylindrical well is being drilled into the ground. The well has a diameter of $36\text{ inches}$ and a target depth of $70\text{ feet}$ . If soil is extracted from the well at a constant rate of $11\text{ cubic feet per hour}$ , how many hours will it take to drill the well completely? (Use $\frac{22}{7}$ for $\pi$ )

Cevabı ve açıklamayı göster

Cevap: 45

Cevap

45

The correct answer is found by first ensuring all measurements are in feet. A diameter of

36\text{ inches}

is equal to

3\text{ feet}

, making the radius

1.5\text{ feet}

(or

\frac{3}{2}\text{ feet}

). Using the formula

V = \pi r^2 h

, the total volume of soil is

\frac{22}{7} \times \left(\frac{3}{2}\right)^2 \times 70 = 495\text{ cubic feet}

. Finally, dividing the total volume by the extraction rate of

11\text{ cubic feet per hour}

gives exactly

45\text{ hours}

.

Adım Adım Çözüm

1

Convert the diameter to feet

36\text{ inches} = 3\text{ feet}

Units must be consistent (feet) with the depth and the extraction rate before calculating volume.

2

Determine the radius of the well

r = 1.5\text{ feet}

(or

\frac{3}{2}\text{ feet}

)

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

3

Calculate the volume of the cylindrical well

V = \frac{22}{7} \times \left(\frac{3}{2}\right)^2 \times 70 = \frac{22}{7} \times \frac{9}{4} \times 70 = 495\text{ cubic feet}

This determines the total amount of soil that needs to be extracted.

4

Calculate the time required to drill the well

\text{Time} = \frac{495}{11} = 45\text{ hours}

Time is equal to the total volume divided by the rate of extraction.

Anahtar Kavram

Calculating the volume of a cylinder with unit conversions and applying it to a rate problem.

Soru 2Soru →

A cylindrical oil storage tank has a diameter of $20\text{ feet}$ and a height of $5\text{ feet}$ . Using $3.14$ for $\pi$ , what is the total volume of the tank?

157\text{ cubic feet}

314\text{ cubic feet}

1,570\text{ cubic feet}

6,280\text{ cubic feet}

Cevabı ve açıklamayı göster

Cevap: $1,570\text{ cubic feet}$

Cevap

The total volume of the tank is

1,570\text{ cubic feet}

.

The correct answer is found by first halving the given diameter (20 feet) to get a radius of 10 feet. The radius is then squared (100) and multiplied by the height (5) and pi (3.14) according to the volume formula

V = \pi r^2 h

. The calculation

3.14 \times 100 \times 5

equals

1,570\text{ cubic feet}

.

Adım Adım Çözüm

1

Find the radius of the cylindrical tank.

r = \frac{d}{2} = \frac{20}{2} = 10\text{ feet}

The volume formula for a cylinder requires the radius, not the diameter.

2

Set up the volume formula for a cylinder.

V = \pi r^2 h

This is the standard geometric formula to find the volume of any cylinder.

3

Substitute the known values into the formula and solve.

V = 3.14 \times (10)^2 \times 5 = 3.14 \times 100 \times 5 = 314 \times 5 = 1,570\text{ cubic feet}

Squaring 10 gives 100, which moves the decimal point in 3.14 two places to the right, making the final multiplication straightforward.

Anahtar Kavram

Calculating the volume of a cylinder using the radius and height.

Soru 3Soru →

A cylindrical farm silo is built to store corn. The interior of the silo has a radius of $7\text{ feet}$ and a height of $20\text{ feet}$ . Using $\frac{22}{7}$ for $\pi$ , what is the total volume of the silo in cubic feet?

Cevabı ve açıklamayı göster

Cevap: 3080

Cevap

The total volume of the silo is 3080 cubic feet.

The correct answer is calculated by applying the volume formula for a cylinder,

V = \pi r^2 h

. Substituting

7

for the radius and

20

for the height, along with

\frac{22}{7}

for

\pi

, results in

3080\text{ cubic feet}

.

Adım Adım Çözüm

1

Recall the volume formula

V = \pi r^2 h

This is the standard formula required to find the volume of any cylinder.

2

Substitute the given values

V = \frac{22}{7} \times (7)^2 \times 20

We plug in

r = 7

,

h = 20

, and

\pi = \frac{22}{7}

as provided in the question.

3

Square the radius

V = \frac{22}{7} \times 49 \times 20

Following the order of operations, exponents must be calculated before multiplication.

4

Multiply and simplify

V = 22 \times 7 \times 20 = 3080

Simplifying the fraction

\frac{22}{7} \times 49

leaves

22 \times 7 = 154

. Multiplying

154

by

20

yields

3080

.

Anahtar Kavram

Calculating the volume of a cylinder

Soru 4Soru →

A city construction crew is installing a new cylindrical concrete drainage pipe. The interior of the pipe has a diameter of $10\text{ feet}$ and a length of $100\text{ feet}$ . Using $3.14$ for $\pi$ , what is the maximum volume of water the pipe can hold when completely full, in cubic feet?

Cevabı ve açıklamayı göster

Cevap: 7850

Cevap

The pipe can hold a maximum volume of 7,850 cubic feet.

The volume of a cylinder is found by multiplying the area of its circular base by its height or length (

V = \pi r^2 h

). First, convert the given diameter of

10\text{ feet}

to a radius by dividing by

2

, yielding a radius of

5\text{ feet}

. Next, square the radius to get

25

. Multiply this by the length (

100\text{ feet}

) and the approximate value of

\pi

(

3.14

).

3.14 \times 25 \times 100 = 7,850\text{ cubic feet}

.

Adım Adım Çözüm

1

Find the radius of the pipe.

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

The volume formula for a cylinder requires the radius, which is half of the diameter.

2

Apply the volume formula for a cylinder.

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

To find the capacity of the cylindrical pipe, we must calculate its three-dimensional space, which is volume.

3

Substitute the known values into the formula.

V = 3.14 \times 5^2 \times 100

We use the given values: radius

5\text{ feet}

, height (or length)

100\text{ feet}

, and

\pi \approx 3.14

.

4

Calculate the final volume.

V = 3.14 \times 25 \times 100 = 3.14 \times 2500 = 7850

Squaring the radius gives

25

. Multiplying by

100

and

3.14

gives the total volume in cubic feet.

Anahtar Kavram

Calculating the volume of a cylinder using the formula

V = \pi r^2 h

, ensuring the given diameter is converted to a radius before performing calculations.

Soru 5Soru →

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

Cevabı ve açıklamayı göster

Cevap: 15

Cevap

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}

.

Adım Adım Çözüm

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.

Anahtar Kavram

Cylindrical Volume Applications

Soru 6Soru →

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?

Cevabı ve açıklamayı göster

Cevap: 22

Cevap

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}

.

Adım Adım Çözüm

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.

Anahtar Kavram

Cylindrical Volume Applications

Cylindrical Volume Applications

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