What mass of butter, which has a usable energy content of `6.0 KCal //g ( = 6000 cal //g )` , would be equivalent to the change in gravitational potential energy of a 78.0 kg man who ascends from sea level to the top of Mt. Everset, at elevation 8.84 km? Assume that the average g for the ascent is `9.80 m//s^(2)`.

Answer

To find the mass of butter equivalent to the change in gravitational potential energy of a 78.0 kg man ascending to the top of Mt. Everest, we can follow these steps:

### Step 1: Calculate the change in gravitational potential energy (PE)
The formula for gravitational potential energy is given by:
\[ PE = m \cdot g \cdot h \]
Where:
- \( m \) = mass of the man = 78.0 kg
- \( g \) = acceleration due to gravity = 9.80 m/s²
- \( h \) = height of Mt. Everest = 8.84 km = 8840 m

Substituting the values:
\[ PE = 78.0 \, \text{kg} \cdot 9.80 \, \text{m/s}^2 \cdot 8840 \, \text{m} \]

### Step 2: Perform the calculation
Calculating the potential energy:
\[ PE = 78.0 \cdot 9.80 \cdot 8840 \]
\[ PE = 78.0 \cdot 9.80 = 764.4 \, \text{kg m/s}^2 \]
\[ PE = 764.4 \cdot 8840 = 676,032 \, \text{J} \]

### Step 3: Convert the energy content of butter to Joules
The energy content of butter is given as \( 6.0 \, \text{KCal/g} \). We need to convert this to Joules:
\[ 1 \, \text{KCal} = 4184 \, \text{J} \]
Thus,
\[ 6.0 \, \text{KCal/g} = 6.0 \cdot 4184 \, \text{J/g} = 25,104 \, \text{J/g} \]

### Step 4: Calculate the mass of butter required
Now, we can find the mass of butter required to provide the same amount of energy as the potential energy calculated earlier. We use the formula:
\[ \text{mass of butter} = \frac{\text{PE}}{\text{energy content per gram}} \]
Substituting the values:
\[ \text{mass of butter} = \frac{676032 \, \text{J}}{25104 \, \text{J/g}} \]

### Step 5: Perform the final calculation
Calculating the mass of butter:
\[ \text{mass of butter} = \frac{676032}{25104} \approx 26.9 \, \text{g} \]

### Conclusion
The mass of butter equivalent to the change in gravitational potential energy of the man is approximately **26.9 grams**.