# Cups A and B are cone shaped and have heights of 32 cm and 27 cm and openings with radii of 12 cm and 13 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?

Jan 22, 2018

$31.9$ cm high of cup A will be filled.

#### Explanation:

Height and radius of cup A is ${h}_{a} = 32 c m , {r}_{a} = 12$ cm

Volume om cone is $\frac{1}{3} \cdot \pi \cdot {r}^{2} \cdot h \therefore {V}_{a} = \frac{1}{3} \cdot \pi \cdot {12}^{2} \cdot 32$ or

${V}_{a} = 4825.49$ cubic cm.

Height and radius of cup B is ${h}_{b} = 27 c m , {r}_{b} = 13$ cm

$\therefore {V}_{b} = \frac{1}{3} \cdot \pi \cdot {13}^{2} \cdot 27 \mathmr{and} {V}_{b} = 4778.36$ cubic cm.

Since ${V}_{a} > {V}_{b}$ , the content will not overflow.

The ratio of radius and hight of cup A is

$\frac{r}{h} = \frac{12}{32} = \frac{3}{8}$. The ratio of radius and hight of water cone

is ${r}_{w} / {h}_{w} = \frac{3}{8} \mathmr{and} {r}_{w} = \frac{3 \cdot {h}_{w}}{8}$

. The volume of water cone is ${V}_{w} = 4778.36$ cubic cm.

$\therefore \frac{1}{3} \cdot \pi \cdot {r}_{w}^{2} \cdot {h}_{w} = 4778.36$ or

$\frac{1}{3} \cdot \pi \cdot {\left(\frac{3 \cdot {h}_{w}}{8}\right)}^{2} \cdot {h}_{w} = 4778.36$ or

${h}_{w}^{3} = \frac{4778.36 \cdot 64}{3 \cdot \pi} \approx 32447.9835$ or

${h}_{w} = \sqrt[3]{32447.9835} \approx 31.90 \left(2 \mathrm{dp}\right)$ cm

$31.9$ cm high of cup A will be filled [Ans]