# Cups A and B are cone shaped and have heights of 25 cm and 18 cm and openings with radii of 8 cm and 9 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 22, 2018

The cup will not overflow and $24.24$ cm high of cup A will be filled.

#### Explanation:

Height and radius of cup A is ${h}_{a} = 25 c m , {r}_{a} = 8$ cm

Volume of cup A is $\frac{1}{3} \cdot \pi \cdot {r}^{2} \cdot h \therefore {V}_{a} = \frac{1}{3} \cdot \pi \cdot {8}^{2} \cdot 25$ or

${V}_{a} \approx 1675.52$ cubic cm.

Height and radius of cup B is ${h}_{b} = 18$ cm ,${r}_{b} = 9$ cm

$\therefore {V}_{b} = \frac{1}{3} \cdot \pi \cdot {9}^{2} \cdot 18 \mathmr{and} {V}_{b} \approx 1526.81$ cubic cm

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

The ratio of radius and height of cup A is $\frac{r}{h} = \frac{8}{25}$.

The ratio of radius and height of content cone in cup A

is ${r}_{c} / {h}_{c} = \frac{8}{25} \mathmr{and} {r}_{c} = \frac{8 \cdot {h}_{c}}{25}$

The volume of content cone is ${V}_{c} \approx 1526.81$ cubic cm.

$\therefore \frac{1}{3} \cdot \pi \cdot {r}_{c}^{2} \cdot {h}_{c} \approx 1526.81$ or

$\frac{1}{3} \cdot \pi \cdot {\left(\frac{8 \cdot {h}_{c}}{25}\right)}^{2} \cdot {h}_{c} \approx 1526.81$ or

${h}_{c}^{3} = \frac{1526.81 \cdot 75}{8 \cdot \pi} \approx 14238.24$ or

${h}_{w} = \sqrt[3]{14238.24} \approx 24.237 \left(2 \mathrm{dp}\right)$ cm

$24.24$ cm high of cup A will be filled [Ans]