A cylindrical well is being drilled into the ground. The well has a diameter of 36\text{ i… — Armed Services Vocational Aptitude Battery (ASVAB)

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 $\pi$ )

Answer: 45 hours

Answer

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}

Step-by-Step Solution

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.

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.

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.

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.

Key Concept

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

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