2021 AMC 12B Problems/Problem 6
Problem
An inverted cone with base radius and height
is full of water. The water is poured into a tall cylinder whose horizontal base has radius of
. What is the height in centimeters of the water in the cylinder?
Solution
The volume of a cone is $\frac{1}{3}\pir^2h$ (Error compiling LaTeX. Unknown error_msg) where is the base radius and
is the height. The water completely fills up the cone so the volume of the water is
.
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 and
.
So the answer is