A man is walking at the rate of 6.5 km/hr towards the foot of a tower…

The correct answer is Option (1) → 2.5 km/hr

Let \( x \) be his distance from the foot of the tower and \( s \) his distance from the top. Then \( s^{2}=x^{2}+120^{2} \).

Differentiating with respect to time:

    \( \displaystyle 2s\frac{ds}{dt}=2x\frac{dx}{dt} \)

    \( \displaystyle \frac{ds}{dt}=\frac{x}{s}\cdot\frac{dx}{dt} \)

When \( x=50 \), \( s=\sqrt{50^{2}+120^{2}}=130 \). He walks towards the foot, so \( \frac{dx}{dt}=-6.5 \):

    \( \displaystyle \frac{ds}{dt}=\frac{50}{130}\times(-6.5)=-2.5\ \text{km/hr} \)

So he approaches the top at 2.5 km/hr.