# Cups A and B are cone shaped and have heights of 32 cm and 16 cm and openings with radii of 15 cm and 12 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 20, 2017

Cup A will not overflow

$4.76 c m \text{ }$high from the bottom of the cup

#### Explanation:

${V}_{\text{cone}} = \frac{1}{3} \pi {r}^{2} h$

where$\text{ "r="the base radius of the cone } ,$

$\text{and "h="the perpendicular height of the cone}$

we have cups a & B

Cup A$\text{ } r = 15 c m , h = 32 c m$

Cup B$\text{ } r = 12 c m , h = 16 c m$

first we find the two volumes

${V}_{A} = \frac{1}{3} \pi {15}^{2} \times 32$

${V}_{A} = 2400 \pi c {m}^{3}$

${V}_{B} = \frac{1}{3} \pi {12}^{2} \times 16$
${V}_{B} = 768 \pi c {m}^{3}$

now cup B is full and poured into cup A

we see

${V}_{B} < {V}_{A}$

so cup A will not overflow

how high up will A be filled.

using the volume formula once more

${V}_{A '} = \frac{1}{3} \pi {r}^{2} h$

where $h$ is the height from the base of the cone

that is from the top of the cup

$768 \cancel{\pi} = \frac{1}{3} \cancel{\pi} {15}^{2} h$

$h = \frac{768 \times 3}{15} ^ 2 = 10.24 c m$

from the bottom of the cup

$H = 15 - 10.24 = 4.76 c m$