# Cups A and B are cone shaped and have heights of 35 cm and 23 cm and openings with radii of 14 cm and 9 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?

Aug 13, 2016

cup $A$ will not overflow.

$= 28.5$ cm

#### Explanation:

Total volume of cup $B = \pi {r}^{2} \frac{h}{3} = \pi {\left(9\right)}^{2} \left(\frac{23}{3}\right) = 621 \pi$
Total volume of cup $A = \pi {r}^{2} \frac{h}{3} = \pi {\left(14\right)}^{2} \left(\frac{35}{3}\right) = 2286.67 \pi$
Since volume of cup $B$ is less than volume of cup $A$; Contents of cup $B$ if poured into cup $A$ will not make cup $A$ overflow
hence cup $A$ will not overflow.

So we have to find H=? the height in Cup $A$ which will have content of same volume as volume of Cup$B$

$\pi {\left(14\right)}^{2} \left(\frac{H}{3}\right) = \pi {\left(9\right)}^{2} \left(23\right)$
or

$\frac{H}{3} = \left({9}^{2} / {14}^{2}\right) \left(23\right) = 9.5$
or

$H = 28.5$ cm