A hemispherical bowl of internal radius $18 ext{ cm}$ is full of liquid. This liquid is to be filled in cylindrical bottles each of radius $3 ext{ cm}$ and height $6 ext{ cm}$. How many bottles are required to empty the bowl?

Question

A hemispherical bowl of internal radius $18 	ext{ cm}$ is full of liquid. This liquid is to be filled in cylindrical bottles each of radius $3 	ext{ cm}$ and height $6 	ext{ cm}$. How many bottles are required to empty the bowl?

Accepted Answer

**Solution:**

1. Calculate the volume of the hemispherical bowl:
   - Volume of a sphere: $V_{sphere} = \frac{4}{3} \pi r^3$
   - Volume of hemisphere: half of the sphere's volume
   - Given radius of bowl $r_{bowl} = 18 	ext{ cm}$

$$
   V_{bowl} = \frac{1}{2} 	imes \frac{4}{3} \pi (18)^3 = \frac{2}{3} \pi (5832) = 3888\pi 	ext{ cm}^3
   $$

2. Calculate the volume of a single cylindrical bottle:
   - Volume of cylinder: $V_{cylinder} = \pi r^2 h$
   - Given radius $r_{bottle} = 3 	ext{ cm}$ and height $h_{bottle} = 6 	ext{ cm}$

$$
   V_{bottle} = \pi (3)^2 (6) = \pi (9)(6) = 54\pi 	ext{ cm}^3
   $$

3. Determine the number of bottles required:
   - Number of bottles = $\frac{V_{bowl}}{V_{bottle}} = \frac{3888\pi}{54\pi} = \frac{3888}{54} = 72$

**Answer:** The number of bottles required to empty the bowl is **72**.

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