Cups A and B are cone shaped and have heights of 35 cm and 29 cm and openings with radii of 12 cm and 16 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 8, 2016

$= 31.42 c m$

Explanation:

Volume of Cup $A = \pi {r}^{2} \left(\frac{h}{3}\right)$ where $r = 12 c m$ and $h = 35 c m$
$= \pi \left({12}^{2}\right) \left(\frac{35}{3}\right) = 1680 \pi$
Volume of Cup $B = \pi {r}^{2} \left(\frac{h}{3}\right)$ where $r = 16 c m$ and $h = 29 c m$
$= \pi \left({16}^{2}\right) \left(\frac{29}{3}\right) = 1508 \pi$
Since volume of $B$ is less than volume of $A$ it will not overflow
Now we can write
$\pi {\left(12\right)}^{2} \left(\frac{H}{3}\right) = 1508 \pi$
or
$H = \frac{4524}{144} = 31.42 c m$
where $H$ is the height of Cup $A$ will be filled.