Question

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

• A. 2
• B. 4
• C. 16
• D. 2.4

Hint:

Remember to simplify ratios carefully and square both sides when dealing with square root equations.

Answer 1

The Correct Option is C

To find the value of \(x\), we need to analyze the given problem using physics concepts related to power. We have two motors capable of raising water from a well, and their power ratio is given by:

\(\frac{3\sqrt{x}}{\sqrt{x}+1}\)

We are given the following details:

• The first motor raises 300 kg of water in 5 minutes from a depth of 100 m.
• The second motor raises 50 kg of water in 2 minutes from the same depth of 100 m.

Let's calculate the power of each motor.

Step 1: Calculate the potential energy (PE) gained by water for each motor

The potential energy gained by water is given by:

PE = m \times g \times h

where:

• m is the mass in kg,
• g is the acceleration due to gravity (approximately 9.8 m/s²),
• h is the height raised.

Step 2: Calculate the power for each motor

Power is given by:

\text{Power} = \frac{\text{Work done}}{\text{Time}} = \frac{m \times g \times h}{t}

Motor 1:

• m_1 = 300 \, \text{kg}
• t_1 = 5 \, \text{minutes} = 300 \, \text{seconds}

Power of Motor 1:

P_1 = \frac{300 \times 9.8 \times 100}{300} = 980 \, \text{W}

Motor 2:

• m_2 = 50 \, \text{kg}
• t_2 = 2 \, \text{minutes} = 120 \, \text{seconds}

Power of Motor 2:

P_2 = \frac{50 \times 9.8 \times 100}{120} = \frac{49000}{120} = 408.33 \, \text{W} \approx 408 \, \text{W}

Step 3: Calculate the ratio of the powers of the two motors

The ratio of powers is:

\frac{P_1}{P_2} = \frac{980}{408} = \frac{245}{102} = \frac{122.5}{51}

Now, equate this ratio to the given expression:

\frac{3\sqrt{x}}{\sqrt{x}+1} = \frac{245}{102}

Step 4: Solve for \(x\)

Cross-multiplying gives:

3\sqrt{x} \cdot 102 = 245 \cdot (\sqrt{x} + 1)

Simplifying:

306\sqrt{x} = 245\sqrt{x} + 245

Subtracting 245\sqrt{x} from both sides:

61\sqrt{x} = 245

Dividing both sides by 61:

\sqrt{x} = \frac{245}{61}

Square both sides to find \(x\):

x = \left(\frac{245}{61}\right)^2 \approx 16

Thus, the value of \(x\) is 16.

The correct answer is 16.