The radius of a cylindrical cistern is $10 \text{ metres}$ and its height is $15 \text{ metres}$. Initially the cistern is empty. We start filling the cistern with water through a pipe whose diameter is $50 \text{ cm}$. Water is coming out of the pipe with a velocity of $5 \text{ m/sec}$. How many minutes will it take in filling the cistern with water?

### Concept & Flow Rate Time To find the total time required to fill a tank, divide the total volume of the tank by the volumetric flow rate of the incoming pipe. $$\text{Time} = \frac{\text{Volume of Tank}}{\text{Cross-Sectional Area of Pipe} \times \text{Flow Velocity}}$$ ### Step-by-Step Solution 1. **Calculate the volume of the cylindrical cistern:** - Radius $R = 10 \text{ m}$, Height $H = 15 \text{ m}$. - $V_{\text{cistern}} = \pi R^2 H = \pi (10)^2 (15) = 1500\pi \text{ m}^3$. 2. **Calculate the volumetric flow rate of the pipe:** - Pipe diameter $= 50 \text{ cm} = 0.5 \text{ m} \Rightarrow$ Pipe radius $r = 0.25 \text{ m} = \frac{1}{4} \text{ m}$. - Pipe cross-sectional area $= \pi r^2 = \pi \left(\frac{1}{4}\right)^2 = \frac{\pi}{16} \text{ m}^2$. - Flow velocity $v = 5 \text{ m/s}$. - Flow rate $= \text{Area} \times \text{Velocity} = \left(\frac{\pi}{16}\right) \times 5 = \frac{5\pi}{16} \text{ m}^3\text{/s}$. 3. **Calculate the total time in seconds:** - $\text{Time in seconds} = \frac{1500\pi}{\frac{5\pi}{16}}$ - $\text{Time} = \frac{1500\pi \times 16}{5\pi} = 300 \times 16 = 4800 \text{ seconds}$. 4. **Convert seconds to minutes:** - $\text{Time in minutes} = \frac{4800}{60} = 80 \text{ minutes}$. ### Exam Strategy & Shortcut Keep $\pi$ in your equations as a symbol; do not substitute $\frac{22}{7}$ or $3.14$, because it will perfectly cancel out when you divide the tank volume by the flow rate. ### Common Pitfall A typical error is failing to convert the pipe's diameter from centimeters ($50 \text{ cm}$) to meters ($0.5 \text{ m}$) before plugging it into the area equation, leading to massive magnitude errors. ### Final Answer Therefore, the correct answer is **80**.