The ratio of powers of two motors is `(3sqrtx)/(sqrtx+1), that are capable of raising `300 kg` water in 5 minutes and `50 kg water in 2 minutes respectively from a wellof `100 m deep .The value of X will be

Options

A. `4`
B. `2`
C. `2.4`
D. `16`

Answer

To solve the problem, we need to find the value of \( x \) given the ratio of the powers of two motors and the work done by each motor in raising water from a well.

### Step-by-Step Solution:

1. **Understand the Work Done by Each Motor**:
The work done \( W \) by each motor can be calculated using the formula:
\[ W = mgh \]
where \( m \) is the mass of the water, \( g \) is the acceleration due to gravity (approximately \( 10 \, \text{m/s}^2 \)), and \( h \) is the height (depth of the well).

2. **Calculate Work Done by Motor 1**:
For the first motor, which raises \( 300 \, \text{kg} \) of water in \( 5 \) minutes:
- Convert time to seconds: \( 5 \, \text{minutes} = 5 \times 60 = 300 \, \text{seconds} \)
- Calculate work done:
\[ W_1 = 300 \times 10 \times 100 = 300000 \, \text{J} \]
- Power \( P_1 \) is given by:
\[ P_1 = \frac{W_1}{t} = \frac{300000}{300} = 1000 \, \text{W} \]

3. **Calculate Work Done by Motor 2**:
For the second motor, which raises \( 50 \, \text{kg} \) of water in \( 2 \) minutes:
- Convert time to seconds: \( 2 \, \text{minutes} = 2 \times 60 = 120 \, \text{seconds} \)
- Calculate work done:
\[ W_2 = 50 \times 10 \times 100 = 50000 \, \text{J} \]
- Power \( P_2 \) is given by:
\[ P_2 = \frac{W_2}{t} = \frac{50000}{120} \approx 416.67 \, \text{W} \]

4. **Set Up the Ratio of Powers**:
We are given the ratio of the powers of the two motors:
\[ \frac{P_1}{P_2} = \frac{3\sqrt{x}}{\sqrt{x} + 1} \]
Substituting the values of \( P_1 \) and \( P_2 \):
\[ \frac{1000}{416.67} \approx 2.4 \]

5. **Equate and Solve for \( x \)**:
Set up the equation:
\[ 2.4 = \frac{3\sqrt{x}}{\sqrt{x} + 1} \]
Cross-multiply:
\[ 2.4(\sqrt{x} + 1) = 3\sqrt{x} \]
Expanding:
\[ 2.4\sqrt{x} + 2.4 = 3\sqrt{x} \]
Rearranging gives:
\[ 3\sqrt{x} - 2.4\sqrt{x} = 2.4 \]
Simplifying:
\[ 0.6\sqrt{x} = 2.4 \]
Dividing both sides by \( 0.6 \):
\[ \sqrt{x} = \frac{2.4}{0.6} = 4 \]
Squaring both sides:
\[ x = 16 \]

### Final Answer:
The value of \( x \) is \( 16 \).