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Difficulty: HardVolume and Surface Area of 3D Figures

A cylindrical water tank has a volume of 250π250\pi cubic feet and a height of 1010 feet. The exterior curved side (lateral surface) of the tank needs a new coat of paint, but the flat top and bottom bases will not be painted. What is the total area, in square feet, that needs to be painted?

  1. A
    150π150\pi
  2. B
    200π200\pi
  3. 100π100\piAnswer
  4. D
    250π250\pi

Answer

The total area that needs to be painted is 100π100\pi square feet.
The area to be painted is the lateral surface area of the cylinder, calculated with the formula 2πrh2\pi r h. To find the unknown radius, use the given volume: 250π=πr2(10)250\pi = \pi r^2(10). Solving this yields r2=25r^2 = 25, so r=5r = 5. Substituting r=5r = 5 and h=10h = 10 into the lateral surface area formula gives 2π(5)(10)=100π2\pi(5)(10) = 100\pi square feet.

Step-by-Step Solution

1
Set up the volume formula for a cylinder to solve for the radius.
V=πr2h⇒250π=πr2(10)V = \pi r^2 h \Rightarrow 250\pi = \pi r^2 (10)
The volume and height are given, so the radius is the only unknown variable.
2
Solve for the radius (rr).
250π=10πr2⇒r2=25⇒r=5250\pi = 10\pi r^2 \Rightarrow r^2 = 25 \Rightarrow r = 5 feet
Dividing both sides by 10π10\pi isolates r2r^2, and taking the square root gives the radius.
3
Identify the correct formula for the area to be painted.
Lateral Surface Area = 2πrh2\pi r h
The problem specifies painting only the exterior curved side (lateral surface), excluding the top and bottom bases.
4
Calculate the lateral surface area.
2π(5)(10)=100π2\pi(5)(10) = 100\pi square feet
Substituting r=5r = 5 and h=10h = 10 into the formula yields the final painted area.

Key Concept

Solving for an unknown dimension using the volume formula, then calculating the lateral surface area of a cylinder.
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Topics

Volume and Surface Area of 3D Figures

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