A 60 kg man runs up a staircase in 12 seconds while 50 kg man runs up the same staircase in 11, seconds, the ratio of the rate of doing their work is

Options

A. `6:5`
B. `12:11`
C. `11:10`
D. `10:11`

Answer

To solve the problem, we need to find the ratio of the rate of doing work (power) of two men running up a staircase. The power can be calculated using the formula:

\[ P = \frac{W}{t} \]

where \( W \) is the work done and \( t \) is the time taken. The work done against gravity when a person climbs a height \( h \) is given by:

\[ W = mgh \]

where \( m \) is the mass of the person, \( g \) is the acceleration due to gravity, and \( h \) is the height climbed.

### Step-by-step Solution:

1. **Identify the masses and times:**
- Mass of the first man, \( m_1 = 60 \, \text{kg} \)
- Time taken by the first man, \( t_1 = 12 \, \text{s} \)
- Mass of the second man, \( m_2 = 50 \, \text{kg} \)
- Time taken by the second man, \( t_2 = 11 \, \text{s} \)

2. **Calculate the work done by each man:**
- For the first man:
\[ W_1 = m_1 g h = 60 \, \text{kg} \cdot g \cdot h \]
- For the second man:
\[ W_2 = m_2 g h = 50 \, \text{kg} \cdot g \cdot h \]

3. **Calculate the power for each man:**
- Power of the first man:
\[ P_1 = \frac{W_1}{t_1} = \frac{60 \, g \, h}{12} = 5 \, g \, h \]
- Power of the second man:
\[ P_2 = \frac{W_2}{t_2} = \frac{50 \, g \, h}{11} \approx 4.545 \, g \, h \]

4. **Find the ratio of their powers:**
\[ \text{Ratio} = \frac{P_1}{P_2} = \frac{5 \, g \, h}{4.545 \, g \, h} \]
The \( g \) and \( h \) cancel out:
\[ \text{Ratio} = \frac{5}{4.545} \approx 1.1 \]

5. **Simplifying the ratio:**
To express this in a more manageable form, we can multiply both the numerator and denominator by 11 to avoid decimals:
\[ \text{Ratio} = \frac{5 \times 11}{4.545 \times 11} \approx \frac{55}{50} = \frac{11}{10} \]

### Final Answer:
The ratio of the rate of doing their work is approximately \( \frac{11}{10} \).