From what height must a block of ice be dropped in order that it may melt completely. It is assumed that the whole of energy is retained by ice. Given that latent heat of ice `= 80 cal g^(-1)`.

Answer

To determine the height from which a block of ice must be dropped in order to melt completely, we can use the principle of conservation of energy. The potential energy lost by the ice block when it falls will be equal to the heat energy required to melt the ice.

### Step-by-Step Solution:

1. **Identify the Variables**:
- Let \( m \) be the mass of the ice block (in grams).
- The latent heat of fusion of ice, \( L = 80 \, \text{cal/g} \).
- The acceleration due to gravity, \( g = 9.8 \, \text{m/s}^2 \).

2. **Calculate the Heat Required to Melt the Ice**:
The heat required to melt the ice can be calculated using the formula:
\[ Q = m \cdot L \]
where \( Q \) is the heat energy required.

3. **Calculate the Potential Energy**:
The potential energy (PE) of the ice block at height \( h \) is given by:
\[ PE = m \cdot g \cdot h \]

4. **Set the Potential Energy Equal to the Heat Required**:
According to the conservation of energy, the potential energy lost by the ice block will be equal to the heat energy required to melt it:
\[ m \cdot g \cdot h = m \cdot L \]

5. **Cancel the Mass \( m \)**:
Since \( m \) appears on both sides of the equation, we can cancel it out (assuming \( m \neq 0 \)):
\[ g \cdot h = L \]

6. **Solve for Height \( h \)**:
Rearranging the equation to solve for \( h \):
\[ h = \frac{L}{g} \]

7. **Substitute the Values**:
Now substitute the values of \( L \) and \( g \):
\[ h = \frac{80 \, \text{cal/g}}{9.8 \, \text{m/s}^2} \]

8. **Convert Calories to Joules (if necessary)**:
Since \( 1 \, \text{cal} = 4.184 \, \text{J} \), we can convert the latent heat into joules:
\[ L = 80 \, \text{cal/g} \times 4.184 \, \text{J/cal} = 334.72 \, \text{J/g} \]
Thus,
\[ h = \frac{334.72 \, \text{J/g}}{9.8 \, \text{m/s}^2} \]

9. **Calculate the Height**:
\[ h \approx \frac{334.72}{9.8} \approx 34.14 \, \text{m} \]

### Final Answer:
The block of ice must be dropped from a height of approximately **34.14 meters** to melt completely.