2021 AMC 12B Spring Problem 6

Attempt Problem 6 of the 2021 AMC 12B Spring below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2021 AMC 12B Spring solutions, or check the answer key.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

An inverted cone with base radius

12

cm and height

18

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

24

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

1.5

3

4

4.5

6

The volume of a cone is

\dfrac13\pi r^2 h

and of a cylinder is

\pi r^2 h

Set the cone’s volume equal to

\pi\cdot 24^2\cdot h

and solve for

h

The cone holds

\dfrac13\pi(12)^2(18)=864\pi

cubic centimeters of water.

Poured into the cylinder, this fills to height

h

where

\pi(24)^2 h=864\pi.

Then

576h=864,

h=1.5.

Thus, the correct answer is A.

Problem 6 in other years