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Problem 572

Geometry Difficulty 2.7 Multiple choice CEMC Pascal · Canada · 2025

Two cylinders are standing on a flat table. Cylinder A has radius
$2$ and height $8$ . Cylinder B has radius $8$ and height $2$ . Cylinder A is $\frac{3}{4}$ full of water and Cylinder B is empty.If all of the water from Cylinder A is then poured into Cylinder B, what fraction of Cylinder B is full of water? (The volume of a cylinder with radius $r$ and height $h$ is $\pi r^2h$ .)

Pick one

Official solution

Solution 1:

The volume of Cylinder A is $\pi\times2^2\times8=32\pi$ , and so the volume of water in Cylinder A is $\frac34\times32\pi=24\pi$ . If all of the water from Cylinder A is poured into Cylinder B, the volume of water in Cylinder B is $24\pi$ . The volume of Cylinder B is $\pi\times8^2\times2=128\pi$ , and so the fraction of Cylinder B that is full of water is $\dfrac{24\pi}{128\pi}=\dfrac{3}{16}$ . Solution 2: As in Solution 1, the volume of water in Cylinder B would be $24\pi$ . Suppose that the height of the water in Cylinder B is $h$ . Then, $\pi\times8^2\times h=24\pi$ or $h=\dfrac{24\pi}{64\pi}=\dfrac{3}{8}$ . Since the height of Cylinder B is $2$ and the height of the water is $\dfrac38$ , then the fraction of Cylinder B that is full of water is $\dfrac{3/8}{2}=\dfrac{3}{16}$ .

Figure for this problem

Source: CEMC, University of Waterloo, licensed CC-BY-NC-4.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.