Question

A vertical cylindrical container is filled with oil. A solid hemispherical stone of radius 7 cm is immersed completely, and the oil level rises by 3 cm. What is the radius of the cylinder?

• A. 6.29 cm
• B. 7.43 cm
• C. 8.73 cm
• D. 11.41 cm

Answer 1

Answer:

C. 8.73 cm

Volume displaced by the hemispherical stone equals the rise in oil volume in the cylinder.

Volume of hemisphere

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

Given (r=7) cm:

$V=\frac{2}{3}\pi(7^3)$
$=\frac{2}{3}\pi(343)$
$=\frac{686}{3}\pi$

Rise in cylinder volume

If cylinder radius is (R) and oil rises by (3) cm:

$\text{Volume rise}=\pi R^2(3)$
Equate volumes:

$3\pi R^2=\frac{686}{3}\pi$

Cancel $(\pi):$

$3R^2=\frac{686}{3}$
$9R^2=686$
$R^2=\frac{686}{9}$
$R=\frac{\sqrt{686}}{3}$
$R \approx \frac{26.19}{3}$
$R \approx 8.73 \text{ cm}$

Therefore, the radius of the cylinder is approximately:

$\boxed{8.73 \text{ cm}}$