Question

If a man of height 1.8 m is walking away from the foot of a light pole of height 6 m with a speed of 7 km per hour on a straight horizontal road opposite to the pole, then the rate of change of the length of his shadow is (in kmph):

• A. \( 7 \)
• B. \( 5 \)
• C. \( 3 \)
• D. \( 2 \)

Hint:

For rate-of-change problems involving shadows, use similar triangles to relate distances and differentiate with respect to time.

Answer 1

The Correct Option is C

Step 1: Define Variables
Let: - \( x \) be the distance of the man from the base of the pole,
- \( s \) be the length of his shadow,
- The height of the pole is \( 6 \) m,
- The height of the man is \( 1.8 \) m,
- The man is walking away at \( \frac{7}{\text{kmph}} \).
Step 2: Use Similar Triangles
By similar triangles: \[ \frac{6}{x+s} = \frac{1.8}{s}. \] Cross multiplying: \[ 6s = 1.8(x + s). \] Rearranging: \[ 6s - 1.8s = 1.8x. \] \[ 4.2s = 1.8x. \] \[ s = \frac{1.8}{4.2} x = \frac{3}{7} x. \] Step 3: Differentiate with Respect to Time
Differentiating both sides: \[ \frac{ds}{dt} = \frac{3}{7} \frac{dx}{dt}. \] \[ \frac{ds}{dt} = \frac{3}{7} \times 7. \] \[ \frac{ds}{dt} = 3. \] Step 4: Conclusion
Thus, the correct answer is: \[ \mathbf{3} \text{ kmph}. \]