2021 AMC 12B Problems/Problem 6

Problem

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 radius of $24\mathrm{cm}$ . What is the height in centimeters of the water in the cylinder?

$\textbf{(A)} ~1.5 \qquad\textbf{(B)} ~3 \qquad\textbf{(C)} ~4 \qquad\textbf{(D)} ~4.5 \qquad\textbf{(E)} ~6$

Solution

The volume of a cone is $\frac{1}{3}\pir^2h$ (Error compiling LaTeX. Unknown error_msg) where $r$ is the base radius and $h$ is the height. The water completely fills up the cone so the volume of the water is $\frac{1}{3}\cdot18\cdot144\pi = 6\cdot144\pi$ .

The volume of a cylinder is $\pir^2h$ (Error compiling LaTeX. Unknown error_msg) so the volume of the water in the cylinder would be $24\cdot24\cdot\pi\cdoth$ (Error compiling LaTeX. Unknown error_msg).

We can equate this two equations like this $24\cdot24\cdot\pi\cdoth = 6\cdot144\pi$ (Error compiling LaTeX. Unknown error_msg). We get $4h = 6$ and $h=\frac{6}{4}$ .

So the answer is $1.5 = \boxed{\textbf{(A)}}.$

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2021 AMC 12B Problems/Problem 6

Problem

Solution