# Cups A and B are cone shaped and have heights of 12 cm and 18 cm and openings with radii of 6 cm and 4 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 9, 2018

$10.48$ cm high of cup A will be filled

#### Explanation:

Height and radius of cup A is ${h}_{a} = 12 c m , {r}_{a} = 6$ cm

Volume om cone is $\frac{1}{3} \cdot \pi \cdot {r}^{2} \cdot h \therefore {V}_{a} = \frac{1}{3} \cdot \pi \cdot {6}^{2} \cdot 12$ or

${V}_{a} \approx 452.39$ cubic cm.

Height and radius of cup B is ${h}_{b} = 18$ cm ,${r}_{b} = 4$ cm

$\therefore {V}_{b} = \frac{1}{3} \cdot \pi \cdot {4}^{2} \cdot 18 \mathmr{and} {V}_{b} \approx 301.59$ cubic cm.

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

The ratio of radius and hight of cup A is

$\frac{r}{h} = \frac{6}{12} = \frac{1}{2}$. The ratio of radius and hight of water cone

is ${r}_{w} / {h}_{w} = \frac{1}{2} \mathmr{and} {r}_{w} = {h}_{w} / 2$

The volume of water cone is ${V}_{w} = 301.59$ cubic cm.

$\therefore \frac{1}{3} \cdot \pi \cdot {r}_{w}^{2} \cdot {h}_{w} = 301.59$ or

$\frac{1}{3} \cdot \pi \cdot {\left({h}_{w} / 2\right)}^{2} \cdot {h}_{w} = 301.59$ or

${h}_{w}^{3} = \frac{301.59 \cdot 12}{\pi} \approx 1151.99$ or

${h}_{w} = \sqrt[3]{1151.99} \approx 10.48 \left(2 \mathrm{dp}\right)$ cm

$10.48$ cm high of cup A will be filled [Ans]