The ratio of powers of two motors is \(\frac{3\sqrt x}{\sqrt x+1}\), that are capable of raising 300 kg water in 5 minutes and 50 kg water in 2 minutes respectively from a well of 100 m deep. The value of x will be
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Remember to simplify ratios carefully and square both sides when dealing with square root equations.
- 2
- 4
- 16
- 2.4
The Correct Option is C
Solution and Explanation
The power \( P \) of a motor is given by: \[ P = \frac{mgh}{t}, \] where \( m \) is the mass of water raised, \( g \) is the acceleration due to gravity, \( h \) is the height, and \( t \) is the time taken.
For the two motors: \[ P_1 = \frac{300 \cdot g \cdot 100}{5 \cdot 60}, \quad P_2 = \frac{50 \cdot g \cdot 100}{2 \cdot 60}. \]
Simplify the powers: \[ P_1 = \frac{300 \cdot 100}{5 \cdot 60}, \quad P_2 = \frac{50 \cdot 100}{2 \cdot 60}. \]
\[ P_1 = 100, \quad P_2 = \frac{250}{6}. \]
The ratio of powers is given by: \[ \frac{P_1}{P_2} = \frac{100}{\frac{250}{6}} = \frac{100 \cdot 6}{250} = \frac{600}{250} = \frac{12}{5}. \]
Equating this with the given ratio: \[ \frac{3\sqrt{x}}{\sqrt{x+1}} = \frac{12}{5}. \]
Cross-multiply: \[ 5 \cdot 3\sqrt{x} = 12 \cdot \sqrt{x+1}. \]
Simplify: \[ 15\sqrt{x} = 12\sqrt{x+1}. \]
Square both sides: \[ 225x = 144(x+1). \]
Expand and simplify: \[ 225x = 144x + 144. \]
\[ 225x - 144x = 144. \]
\[ 81x = 144 \quad \Rightarrow \quad x = \frac{144}{81} = 16. \]
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