A steel wire of length `4.7 m` and cross-sectional area `3 xx 10^(-6) m^(2)` stretches by the same amount as a copper wire of length `3.5 m` and cross-sectional area of `4 xx 10^(-6) m^(2)` under a given load. The ratio of Young's modulus of steel to that of copper is

Options

A. `1.8`
B. `3.6`
C. `0.6`
D. `8.7`

Explanation

To solve the problem, we need to find the ratio of Young's modulus of steel (Ys) to that of copper (Yc) given the lengths and cross-sectional areas of both wires.

### Step-by-Step Solution:

1. **Understanding Young's Modulus**:
Young's modulus (Y) is defined as the ratio of stress to strain. The formula for Young's modulus is given by:
\[ Y = \frac{F \cdot L}{A \cdot \Delta L} \]
where:
- \( F \) = force applied,
- \( L \) = original length of the wire,
- \( A \) = cross-sectional area,
- \( \Delta L \) = change in length (elongation).

2. **Given Information**:
- For the steel wire:
- Length (\( L_s \)) = 4.7 m
- Cross-sectional area (\( A_s \)) = \( 3 \times 10^{-6} \, m^2 \)
- For the copper wire:
- Length (\( L_c \)) = 3.5 m
- Cross-sectional area (\( A_c \)) = \( 4 \times 10^{-6} \, m^2 \)
- Both wires stretch by the same amount (\( \Delta L \)) under the same load.

3. **Setting Up the Ratio**:
Since both wires are subjected to the same load and have the same elongation, we can express the ratio of Young's moduli as:
\[ \frac{Y_s}{Y_c} = \frac{L_s / A_s}{L_c / A_c} \]
This can be rearranged to:
\[ \frac{Y_s}{Y_c} = \frac{L_s \cdot A_c}{L_c \cdot A_s} \]

4. **Substituting Values**:
Now we can substitute the known values into the equation:
\[ \frac{Y_s}{Y_c} = \frac{4.7 \cdot (4 \times 10^{-6})}{3.5 \cdot (3 \times 10^{-6})} \]

5. **Simplifying the Expression**:
The \( 10^{-6} \) terms cancel out:
\[ \frac{Y_s}{Y_c} = \frac{4.7 \cdot 4}{3.5 \cdot 3} \]

6. **Calculating the Numerator and Denominator**:
- Numerator: \( 4.7 \cdot 4 = 18.8 \)
- Denominator: \( 3.5 \cdot 3 = 10.5 \)

7. **Final Calculation**:
Now we can calculate the ratio:
\[ \frac{Y_s}{Y_c} = \frac{18.8}{10.5} \approx 1.7905 \approx 1.8 \]

### Conclusion:
The ratio of Young's modulus of steel to that of copper is approximately:
\[ \frac{Y_s}{Y_c} \approx 1.8 \]