A man grows into a giant such that his linear dimension increase by a factor of 9. Assuming that his density remains same, the stress in the leg will change by a factor of

Options

A. 81
B. `(1)/(81)`
C. 9
D. `(1)/(9)`

Explanation

To solve the problem, we need to determine how the stress in the leg of a man changes when his linear dimensions increase by a factor of 9, while maintaining the same density.

### Step-by-Step Solution:

1. **Understanding Stress**:
Stress (S) is defined as the force (F) applied per unit area (A). Mathematically, it is expressed as:
\[ S = \frac{F}{A} \]

2. **Determining the Force**:
The force acting on the man's legs is due to his weight (W), which can be expressed as:
\[ W = m \cdot g \]
where \(m\) is the mass and \(g\) is the acceleration due to gravity.

3. **Finding the Mass**:
The mass \(m\) can be expressed in terms of volume (V) and density (\(\rho\)):
\[ m = V \cdot \rho \]

4. **Volume Calculation**:
The volume of a man, assuming he is roughly a three-dimensional object, is proportional to the cube of his linear dimensions. If the linear dimensions increase by a factor of 9, then:
\[ V_2 = 9^3 \cdot V_1 = 729 \cdot V_1 \]

5. **Area Calculation**:
The cross-sectional area (A) of the man's legs is proportional to the square of his linear dimensions. Therefore, if the linear dimensions increase by a factor of 9:
\[ A_2 = 9^2 \cdot A_1 = 81 \cdot A_1 \]

6. **Calculating Stress**:
Now we can express the initial and final stress:
- Initial stress \(S_1\):
\[ S_1 = \frac{W_1}{A_1} = \frac{m_1 \cdot g}{A_1} = \frac{V_1 \cdot \rho \cdot g}{A_1} \]
- Final stress \(S_2\):
\[ S_2 = \frac{W_2}{A_2} = \frac{m_2 \cdot g}{A_2} = \frac{V_2 \cdot \rho \cdot g}{A_2} = \frac{(729 \cdot V_1) \cdot \rho \cdot g}{(81 \cdot A_1)} \]

7. **Finding the Ratio of Stresses**:
Now, we can find the ratio of the final stress to the initial stress:
\[ \frac{S_2}{S_1} = \frac{(729 \cdot V_1) \cdot \rho \cdot g / (81 \cdot A_1)}{(V_1 \cdot \rho \cdot g / A_1)} = \frac{729}{81} = 9 \]

8. **Conclusion**:
Therefore, the stress in the leg will change by a factor of 9.

### Final Answer:
The stress in the leg will change by a factor of **9**.