# Cups A and B are cone shaped and have heights of 34 cm and 27 cm and openings with radii of 11 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?

Jun 12, 2018

Cup A will not overflow. It will be filled up to $\textcolor{v i o \le t}{5.58 c m}$ height

#### Explanation:

Cone A ${r}_{a} = 11 c m , {h}_{a} = 34 c m$

Cone B ${r}_{b} = 5 c m , {h}_{b} = 27 c m$

Volume of Cone A ${V}_{a} = \left(\frac{1}{3}\right) \pi {\left({r}_{a}\right)}^{2} {h}_{a}$

${V}_{a} = \left(\frac{1}{3}\right) \pi \cdot {11}^{2} \cdot 34 = 4308.17 c {m}^{3}$

Volume of Cone A ${V}_{b} = \left(\frac{1}{3}\right) \pi {\left({r}_{b}\right)}^{2} {h}_{b}$

${V}_{a} = \left(\frac{1}{3}\right) \pi \cdot {5}^{2} \cdot 27 = 706.86 c {m}^{3}$

Since ${V}_{b} < {V}_{a}$, it will not overflow.

$\left(\frac{1}{3}\right) \pi {\left({r}_{a}\right)}^{2} \cdot h = {V}_{b} = 225 \pi$

$h = \frac{3 \cdot 225 \cdot \pi}{\pi \cdot {11}^{2}} = 5.58 c m$