A man having height 169 cm is standing near a pole. He casts a shadow 130 cm long. What is the length of the pole if it gives a shadow 420 cm long?

Options

A. 550 cm
B. 589 cm
C. 232 cm
D. 546 cm

Answer

To find the length of the pole based on the given information, we can use the concept of similar triangles. The height of the man and the length of his shadow form one triangle, while the height of the pole and the length of its shadow form another triangle.

### Step-by-Step Solution:

1. **Identify the given values**:
- Height of the man (H_m) = 169 cm
- Length of the man's shadow (S_m) = 130 cm
- Length of the pole's shadow (S_p) = 420 cm
- Height of the pole (H_p) = ?

2. **Set up the proportion using similar triangles**:
The triangles formed by the man and his shadow and the pole and its shadow are similar. Therefore, we can set up the following proportion:
\[ \frac{H_m}{S_m} = \frac{H_p}{S_p} \]

3. **Substitute the known values into the proportion**:
\[ \frac{169}{130} = \frac{H_p}{420} \]

4. **Cross-multiply to solve for the height of the pole (H_p)**:
\[ 169 \times 420 = 130 \times H_p \]

5. **Calculate the left side**:
\[ 169 \times 420 = 70980 \]

6. **Now, solve for H_p**:
\[ H_p = \frac{70980}{130} \]

7. **Perform the division**:
\[ H_p = 546 \]

8. **Conclusion**:
The height of the pole is **546 cm**.