Question

A hemispherical bowl is 264 cm round at the brim. Suppose that the bowl is half-full of beverage. If the beverage is served in hemispherical glasses 6 cm in the diameter at the top, how many people would be served?

• A. 1170
• B. 1372
• C. 1270
• D. 1470

Answer 1

Answer:

B. 1372

“264 cm round at the brim” means the circumference of the bowl:

$2\pi R = 264 \Rightarrow R = \frac{264}{2\pi} = \frac{264}{2 \times \frac{22}{7}} = 42 \text{ cm}$

Volume of hemispherical bowl:

$\text{Volume} = \frac{2}{3}\pi R^3 = \frac{2}{3}\pi (42)^3$

Since it is half-full:
$\text{Liquid volume} = \frac{1}{2} \times \frac{2}{3}\pi (42)^3 = \frac{1}{3}\pi (42)^3$

Glass details:

Diameter = 6 cm ⇒ radius = 3 cm

Volume of one hemispherical glass:
$\frac{2}{3}\pi (3)^3 = \frac{2}{3}\pi \cdot 27 = 18\pi$

Number of glasses:

$\frac{\frac{1}{3}\pi (42)^3}{18\pi}$

$= \frac{(42)^3}{54}$

$42^3 = 74088$

$\frac{74088}{54} = 1372$

Final Answer:

1372 people