# Cups A and B are cone shaped and have heights of 33 cm and 27 cm and openings with radii of 13 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?

Cup A will not overflow because ${V}_{b} < {V}_{a}$
Cup A will be filled up to $h = 22.3303$ cm

#### Explanation:

Compute for the volumes of cups A and B first.
The formula for cone

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

For cup A
${V}_{a} = \frac{1}{3} \cdot \pi \cdot {13}^{2} \cdot 33 = 1859 \pi = 5840.220741927$

For cup B
${V}_{b} = \frac{1}{3} \cdot \pi \cdot {8}^{2} \cdot 27 = 576 \pi = 1809.557368128$

We need 2 equations to solve for unknown height with ${V}_{b}$ poured into ${V}_{a}$

For cup A, we need the ratio of Radius to Height:
${r}_{a} / {h}_{a} = \frac{13}{33}$
and ${r}_{a} = \frac{13}{33} {h}_{a}$

Using volume of ${V}_{b} = 576 \pi$

${V}_{b} = \frac{1}{3} \pi \cdot {r}_{a}^{2} \cdot {h}_{a}$

$576 \pi = \frac{1}{3} \cdot \pi \cdot {\left(\frac{13}{33} \cdot {h}_{a}\right)}^{2} \cdot {h}_{a}$

$\frac{576 \cdot 3 \cdot {33}^{2}}{13} ^ 2 = {h}_{a}^{3}$

h_a=root3((576*3*33^2)/13^2

${h}_{a} = 22.3303 \text{ " }$cm

God bless...I hope the explanation is useful.