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Difficulty: MediumVolume and Surface Area of 3D Figures

A solid cylindrical metal rod with a radius of 33 inches and a length of 88 inches is melted down and completely recast into a new cylindrical rod. If the new rod must have a radius of 22 inches and no material is lost, what is its length in inches?

  1. A
    12
  2. B
    72
  3. 18Answer
  4. D
    36

Answer

The length of the new cylindrical rod is 18 inches.
The correct answer is 18. Because no material is lost, the volume of the new rod must strictly equal the volume of the original rod. The original volume is calculated using the formula V=πr2hV = \pi r^2 h, resulting in π(3)2(8)=72π\pi(3)^2(8) = 72\pi. Setting this equal to the new rod's volume expression, 72π=π(2)2h72\pi = \pi(2)^2 h, simplifies the equation to 72π=4πh72\pi = 4\pi h. Dividing both sides by 4π4\pi yields a final length of 18 inches.

Step-by-Step Solution

1
Calculate the volume of the original cylindrical rod.
V1=πr2h=π(3)2(8)=π(9)(8)=72πV_{1} = \pi r^2 h = \pi(3)^2(8) = \pi(9)(8) = 72\pi cubic inches.
Since the rod is being melted down, the total amount of material (volume) remains constant. We first need the total volume of the original rod.
2
Set up the volume expression for the new cylindrical rod.
V2=πr2h=π(2)2h=4πhV_{2} = \pi r^2 h = \pi(2)^2 h = 4\pi h.
The new rod is also a cylinder, but with a different radius (2 inches) and an unknown length (height).
3
Equate the two volumes and solve for the unknown length.
72π=4πh→h=72π4π=1872\pi = 4\pi h \rightarrow h = \frac{72\pi}{4\pi} = 18 inches.
Because no material is lost during recasting, the initial volume must equal the final volume.

Key Concept

Conservation of volume in 3D cylinders
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Topics

Volume and Surface Area of 3D Figures

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