Use power = work / time

For lifting water from a well of depth $h=100\,\text{m}$, the work done is $W = mgh$ So power is $P=\frac{mgh}{t}$

Power of first motor

It raises $300\,\text{kg}$ of water in $5$ minutes.

Convert time: $5\text{ min} = 300\text{ s}$

Hence, $P_1 = \frac{300\cdot g\cdot 100}{300} = 100g$

Power of second motor

It raises $50\,\text{kg}$ of water in $2$ minutes.

Convert time: $2\text{ min} = 120\text{ s}$

Hence, $P_2 = \frac{50\cdot g\cdot 100}{120} = \frac{5000g}{120} = \frac{125g}{3}$

Find the ratio of powers

$\frac{P_1}{P_2} = \frac{100g}{125g/3} = \frac{100\cdot 3}{125} = \frac{12}{5}$

Given, $\frac{P_1}{P_2} = \frac{3\sqrt{x}}{\sqrt{x}+1}$

Therefore, $\frac{3\sqrt{x}}{\sqrt{x}+1} = \frac{12}{5}$

Solve for $x$

Cross-multiplying: $15\sqrt{x} = 12(\sqrt{x}+1)$ $15\sqrt{x} = 12\sqrt{x}+12$ $3\sqrt{x} = 12$ $\sqrt{x} = 4$ $x = 16$

Check options

• A: $16$ ✅
• B: $4$
• C: $2$
• D: $2.4$

So the correct option is A.