A man 1.5 m tall walks away from the foot of a lamp post at the rate of 1.2 m/s. The lamp is 6 m above the ground. The rate at which the length of his shadow is increasing is
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → 0.4 m/s
Let x be the distance of the man from the foot of the lamp post and s the length of his shadow at time t. The lamp, the top of the man's head and the tip of the shadow lie on one line, so the two right triangles are similar:
\( \displaystyle \frac{6}{x + s} = \frac{1.5}{s} \)
\( 6s = 1.5x + 1.5s \)
\( 4.5s = 1.5x \)
\( \displaystyle s = \frac{x}{3} \)
Differentiating with respect to t, with \(\frac{dx}{dt} = 1.2\):
\( \displaystyle \frac{ds}{dt} = \frac{1}{3}\cdot\frac{dx}{dt} = \frac{1.2}{3} \)
\( = 0.4 \) m/s
Why the other options are incorrect
Option (2) 0.3 m/s — Incorrect. This is \(\frac{1.5}{6} \times 1.2\); the shadow satisfies \(s = \frac{1.5x}{6 - 1.5}\), so the factor is \(\frac{1.5}{4.5}\), not \(\frac{1.5}{6}\).
Option (3) 1.6 m/s — Incorrect. This is the speed of the tip of the shadow, \(\frac{d}{dt}(x + s) = 1.2 + 0.4\), not the rate of change of the shadow's length.
Option (4) 0.8 m/s — Incorrect. With \(s = \frac{x}{3}\) the rate is \(\frac{1.2}{3} = 0.4\), not 0.8.