# Cups A and B are cone shaped and have heights of 37 cm and 27 cm and openings with radii of 9 cm and 5 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?

Apr 29, 2018

A is bigger in both dimensions, so will hold the contents of B at height $h$ for ${V}_{B} = \frac{1}{3} \pi {r}_{A}^{2} h$ or

h = {3 V_B}/{pi r_A^2 } = 3 (1/3 pi r_B^2 h_B}/{\pi r_A^2 } = h_B {r_B^2}/r_A^2 = (27) 5^2/9^2= 25/3

#### Explanation:

The volume of a cone of radius $r$ and height $h$ is given by

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

A is bigger than B in both radius and height, so of course B's volume is less and A will not overflow. We have

${V}_{A} = \frac{1}{3} \pi \left({9}^{2}\right) 37 = 999 \pi {\textrm{c m}}^{3}$

${V}_{B} = \frac{1}{3} \pi \left({5}^{2}\right) \left(27\right) = 225 \pi {\textrm{c m}}^{3}$

The height of A after receiving the contents of B is given by

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

$h = \frac{3 {V}_{B}}{\pi {r}_{A}^{2}} = \frac{3 \cdot 225 \pi}{\pi \left({9}^{2}\right)} = \frac{25}{3}$