A hollow cylinder A with radius $14 ext{ cm}$ is completely filled with water. A solid cylinder B with radius $7 ext{ cm}$ is then placed inside cylinder A. The remaining volume of water after placing cylinder B is $9240 ext{ cm}^3$. Both cylinders have the same height.

If there is a leak in cylinder B that empties it at a rate of $0.5 ext{ cm}^3/ ext{sec}$, how long will it take to empty cylinder B? (Give your answer in minutes)

Question

A hollow cylinder A with radius $14 	ext{ cm}$ is completely filled with water. A solid cylinder B with radius $7 	ext{ cm}$ is then placed inside cylinder A. The remaining volume of water after placing cylinder B is $9240 	ext{ cm}^3$. Both cylinders have the same height.

If there is a leak in cylinder B that empties it at a rate of $0.5 	ext{ cm}^3/	ext{sec}$, how long will it take to empty cylinder B? (Give your answer in minutes)

Accepted Answer

**Solution:**

1. Let the height of both cylinders be $h$ cm.

2. Volume of hollow cylinder A is the volume of water plus the volume of solid cylinder B:
   $$	ext{Volume of A} = \pi 	imes 14^2 	imes h$$

3. Volume of solid cylinder B:
   $$\pi 	imes 7^2 	imes h$$

4. Given the remaining water volume after placing B inside A is $9240 	ext{ cm}^3$:
   $$\pi 	imes 14^2 	imes h - \pi 	imes 7^2 	imes h = 9240$$

5. Simplify:
   $$\pi h (14^2 - 7^2) = 9240$$
   $$\pi h (196 - 49) = 9240$$
   $$\pi h 	imes 147 = 9240$$

6. Substitute $\pi = \frac{22}{7}$:
   $$\frac{22}{7} 	imes h 	imes 147 = 9240$$

7. Calculate:
   $$22 	imes h 	imes 21 = 9240$$
   $$462h = 9240$$
   $$h = \frac{9240}{462} = 20 	ext{ cm}$$

8. Now calculate the volume of cylinder B:
   $$\pi 	imes 7^2 	imes 20 = \frac{22}{7} 	imes 49 	imes 20 = 3080 	ext{ cm}^3$$

9. The leak empties cylinder B at $0.5 	ext{ cm}^3/	ext{sec}$, so time to empty:
   $$\frac{3080}{0.5} = 6160 	ext{ seconds}$$

10. Convert seconds to minutes:
    $$\frac{6160}{60} = 102 \frac{2}{3} 	ext{ minutes}$$

**Answer:** 102 $\frac{2}{3}$ minutes

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