# Cups A and B are cone shaped and have heights of 32 cm and 12 cm and openings with radii of 5 cm and 8 cm, respectively. If cup B is full and its contents are poured into cup A, will cup A overflow? If not how high will cup A be filled?

Feb 22, 2016

When cup B is full and its contents are poured into cup A, cup A will not overflow. It will be filled up to a height of $30.72$ cm

#### Explanation:

Volume of a cone whose radius is $r$ and height is $h$ is given by $\frac{1}{3} \pi {r}^{2} h$.

Hence volume of Cup A is $\frac{1}{3} \pi {5}^{2} \cdot 32$ = $\frac{25 \cdot 32}{3} \pi$ = $\frac{800}{3} \pi$
and volume of Cup B is $\frac{1}{3} \pi {8}^{2} \cdot 12$ = $\frac{64 \cdot 12}{3} \pi$ = $\frac{768}{3} \pi$.

As volume of Cup B is less than that of A, when cup B is full and its contents are poured into cup A, cup A will not overflow.

Let Cup A be filled till height $a$, than $\frac{1}{3} \pi {5}^{2} \cdot a$ = $\frac{768}{3} \pi$

or $25 a = 768$ or $a = \frac{768}{25} = 30.72$ cm