Two men with weights in the ratio 4:3 run up a staircase in time in the ratio 12:11. The ratio of power of the first to that of second is

Options

A. `4/3`
B. `12/11`
C. `48/33`
D. `11/9`

Explanation

To find the ratio of power of the first man to that of the second man, we can follow these steps:

### Step 1: Understand the formula for power
Power is defined as the work done per unit time. Mathematically, it is expressed as:
\[ P = \frac{W}{t} \]
where \( P \) is power, \( W \) is work done, and \( t \) is time taken.

### Step 2: Define the work done by each man
The work done by each man to run up the staircase can be expressed as:
\[ W = mgh \]
where \( m \) is the mass (or weight) of the man, \( g \) is the acceleration due to gravity, and \( h \) is the height of the staircase. Since both men run up the same staircase, the height \( h \) is the same for both.

### Step 3: Set up the ratios
Let the weights of the two men be \( W_1 \) and \( W_2 \). According to the problem, the weights are in the ratio:
\[ \frac{W_1}{W_2} = \frac{4}{3} \]

Let the times taken by the two men to run up the staircase be \( t_1 \) and \( t_2 \). The times are in the ratio:
\[ \frac{t_1}{t_2} = \frac{12}{11} \]

### Step 4: Write the expressions for power
The power for each man can be expressed as:
\[ P_1 = \frac{W_1}{t_1} \]
\[ P_2 = \frac{W_2}{t_2} \]

### Step 5: Find the ratio of powers
To find the ratio of the powers \( \frac{P_1}{P_2} \):
\[ \frac{P_1}{P_2} = \frac{W_1/t_1}{W_2/t_2} = \frac{W_1}{W_2} \cdot \frac{t_2}{t_1} \]

### Step 6: Substitute the known ratios
Substituting the ratios we have:
\[ \frac{P_1}{P_2} = \left(\frac{4}{3}\right) \cdot \left(\frac{11}{12}\right) \]

### Step 7: Simplify the expression
Now, we can simplify this:
\[ \frac{P_1}{P_2} = \frac{4 \times 11}{3 \times 12} = \frac{44}{36} = \frac{11}{9} \]

### Conclusion
Thus, the ratio of power of the first man to that of the second man is:
\[ \frac{P_1}{P_2} = \frac{11}{9} \]

### Final Answer
The ratio of power of the first man to that of the second man is \( \frac{11}{9} \).

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