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Surface Areas and Volumes
NCERT Exemplar Ch 11
CBSE_NCERT_EXEMPLAR_CH11
Grade 10

Question:

A hemispherical tank full of water is emptied by a pipe at the rate of $3\dfrac{4}{7}$ litres per second. How much time will it take to empty half the tank, if it is $3\text{ m}$ in diameter?

Step-by-Step Solution

Key Concept: Radius $r = 1.5\text{ m} = 3/2\text{ m}$. Half-volume of tank $= \dfrac{1}{2} \left(\dfrac{2}{3}\pi r^3\right) = \dfrac{1}{3} \pi r^3$. Convert $\text{m}^3$ to litres ($1\text{ m}^3 = 1000\text{ L}$). Rate $= 25/7\text{ L/s}$.
Stepwise Solution:

Half-volume $= \dfrac{1}{3} \times \dfrac{22}{7} \times \left(\dfrac{3}{2}\right)^3 = \dfrac{1}{3} \times \dfrac{22}{7} \times \dfrac{27}{8} = \dfrac{99}{28}\text{ m}^3$. [1.0 Mark]

In litres $= \dfrac{99}{28} \times 1000 = \dfrac{99000}{28}\text{ L}$. [0.5 Mark]

Emptying rate $= 3\dfrac{4}{7} = \dfrac{25}{7}\text{ L/s}$.
Time required $= \dfrac{99000/28}{25/7} = \dfrac{99000}{28} \times \dfrac{7}{25} = \dfrac{99000}{100} = 990\text{ seconds}$. [1.0 Mark]

$990\text{ seconds} = \dfrac{990}{60} = 16.5\text{ minutes}$ (or $16\text{ min } 30\text{ sec}$). [0.5 Mark]

Marking Scheme:

• Finding half-volume in $\text{m}^3 = 99/28\text{ m}^3$: 1.0 Mark
• Converting to litres ($99000/28\text{ L}$): 0.5 Mark
• Calculating time in seconds $= 990\text{ s}$: 1.0 Mark
• Converting to $16.5$ minutes: 0.5 Mark

Correct Answer:
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