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Try this beautiful problem from Mensuration: Problem based on Cylinder from AMC 10A, 2015.

Two right circular cylinders have the same volume. The radius of the second cylinder is $10 \%$ more than the radius of the first. What is the relationship between the heights of the two cylinders?

- (A) The second height is $10 \%$ less than the first.
- (B) The first height is $10 \%$ more than the second.
- (C) The second height is $21 \%$ less than the first.
- (D) The first height is $21 \%$ more than the second.
- (E) The second height is $80 \%$ of the first.

Mensuration

Cylinder

But try the problem first...

Answer: (D) The first height is $21 \%$ more than the second.

Source

Suggested Reading

AMC-10A (2015) Problem 9

Pre College Mathematics

First hint

Let the radius of the first cylinder be $r_{1}$ and the radius of the second cylinder be $r_{2}$. Also, let the height of the first cylinder be $h_{1}$ and the height of the second cylinder be $h_{2}$.

Can you now finish the problem ..........

Second Hint

According to the problem,

$r_{2}=\frac{11 r_{1}}{10}$

$\pi r_{1}^{2} h_{1}=\pi r_{2}^{2} h_{2}$

can you finish the problem........

Third Hint:

$r_{1}^{2} h_{1}=\frac{121 r_{1}^{2}}{100} h_{2} \Rightarrow h_{1}=\frac{121 h_{2}}{100}$

Therefore the Possible answer will be **(D) The first height is $21 \%$ more than the second.**

- https://www.cheenta.com/surface-area-of-cube-amc-10a-2007-problem-21/
- https://www.youtube.com/watch?v=VLyrlx2DWdA&t=20s

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