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Difficulty: MediumCylindrical Volume Applications

A large cylindrical water cistern used for irrigation has a diameter of 10 feet10\text{ feet} and a depth of 6 feet6\text{ feet}. A heavy-duty pump removes water from the completely full cistern at a constant rate of 31.4 cubic feet31.4\text{ cubic feet} per minute. Using 3.143.14 for π\pi, how many minutes will it take to empty the cistern?

  1. A
    3
  2. B
    6
  3. 15Answer
  4. D
    60

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 feetr = 10 / 2 = 5\text{ feet}. The formula for the volume of a cylinder is V=π×r2×hV = \pi \times r^2 \times h. Plugging in our values: V=3.14×52×6=3.14×25×6=3.14×150=471 cubic feetV = 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 cubic feet31.4\text{ cubic feet} of water each minute. Dividing the total volume by the rate gives the time: 471/31.4=15 minutes471 / 31.4 = 15\text{ minutes}.

Step-by-Step Solution

1
Find the radius of the cistern.
r=10/2=5 feetr = 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=π×r2×hV = \pi \times r^2 \times h.
V=3.14×52×6=3.14×25×6=3.14×150=471 cubic feetV = 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 minutesTime = 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
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Topics

Cylindrical Volume Applications

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