2021 AMC 10B Spring Problem 10

Question

An inverted cone with base radius

12 \mathrm{ cm}

and height

18 \mathrm{ cm}

is full of water. The water is poured into a tall cylinder whose horizontal base has a radius of

24 \mathrm{ cm}.

What is the height in centimeters of the water in the cylinder?

1.5

3

4

4.5

6

Set the cone volume equal to the cylinder volume

The cylinder radius is twice the cone radius

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The volumes must be the same since the water is poured from one to another. The volume of the cone is

\frac{r^2h \pi}3 = \frac{12^2\cdot 18 \pi}3 = 864 \pi.

The volume of the cylinder is

r^2h \pi =24^2h \pi = 576h \pi.

This makes

576 h \pi = 864 \pi,

so

h = 1.5.

Thus, the answer is A .

Problem 10 in other years