Each of two cylinders sits on one of their circular faces on a flat surface. Cylinder A, with radius $6 \mathrm{~cm}$ and height $50 \mathrm{~cm}$, is empty. Cylinder B, with radius $8 \mathrm{~cm}$ and height $50 \mathrm{~cm}$, is full of water. After pouring some water from Cylinder B into Cylinder A, the height of the water in both cylinders is the same. What is the height of the water? (The volume of a cylinder with radius $r$ and height $h$ is $\pi r^{2} h$.)
(A) $28.1 \mathrm{~cm}$
(B) $25.0 \mathrm{~cm}$
(C) $32.0 \mathrm{~cm}$
(D) $44.4 \mathrm{~cm}$
(E) $28.6 \mathrm{~cm}$
To begin, the volume of water in Cylinder B is $\pi \times(8 \mathrm{~cm})^{2} \times 50 \mathrm{~cm}=3200 \pi \mathrm{cm}^{3}$.

After some water is poured from Cylinder B into Cylinder A, the total volume of water in the two cylinders will be $3200 \pi \mathrm{cm}^{3}$ (since no water is lost).

Let $h \mathrm{~cm}$ be the height of the water in each of the two cylinders when the height of the water in both cylinders is the same.

At this time, the volume of water in Cylinder B is $\pi \times(8 \mathrm{~cm})^{2} \times h \mathrm{~cm}=64 \pi h \mathrm{~cm}^{3}$.

At this time, the volume of water in Cylinder A is $\pi \times(6 \mathrm{~cm})^{2} \times h \mathrm{~cm}=36 \pi h \mathrm{~cm}^{3}$.

Thus, the total volume of water in the two cylinders is $64 \pi h \mathrm{~cm}^{3}+36 \pi h \mathrm{~cm}^{3}=100 \pi h \mathrm{~cm}^{3}$, and so $100 \pi h=3200 \pi$ or $h=\frac{3200 \pi}{100 \pi}=32$.

When the height of the water in both cylinders is the same, that height is $32 \mathrm{~cm}$.

ANswer: (C)

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