Question

A man \(1.6\mathrm{m}\) high walks at the rate of \(30\mathrm{m/min}\) away from a lamp which is \(4\mathrm{m}\) above ground. How fast is the man's shadow lengthening?

• A. \(22\mathrm{m/min}\)
• B. \(20\mathrm{m/min}\)
• C. \(15\mathrm{m/min}\)
• D. \(25\mathrm{m/min}\)

Hint:

Use similar triangles to relate shadow length to distance from lamp.

Answer 1

The Correct Option is B

To solve this problem, we will use a combination of geometry and calculus to determine the rate at which the man's shadow is lengthening. Let's break down the problem step-by-step:

1. Identify the components in the scenario:
   • The man is \(1.6 \, \mathrm{m}\) tall.
   • The lamp is \(4 \, \mathrm{m}\) above the ground.
   • The man walks away from the lamp at a rate of \(30 \, \mathrm{m/min}\).
2. Use the concept of similar triangles to relate the height of the lamp, the man's height, the length of the shadow, and the distance walked by the man.
   • Let \(x\) be the distance of the man from the lamp, and \(y\) be the length of the man's shadow.
   • The triangles formed by the lamp post, the man's position, and their respective shadows are similar.
   • From the similarity of triangles, we have the relation: 
     \(\frac{1.6}{4} = \frac{y}{x + y}\).
3. Solve the equation for \(y\):
   • Rearrange the equation: 
     \(1.6(x + y) = 4y\).
   • Expand and simplify: 
     \(1.6x + 1.6y = 4y\).
   • Further simplify to: 
     \(1.6x = 4y - 1.6y = 2.4y\).
   • Isolate \(y\): 
     \(y = \frac{1.6x}{2.4} = \frac{2}{3}x\).
4. Differentiate with respect to time to find the rate at which the shadow is lengthening (i.e., \(\frac{dy}{dt}\)):
   • We know: 
     \(y = \frac{2}{3}x\), so differentiate both sides with respect to \(t\): 
     \(\frac{dy}{dt} = \frac{2}{3}\frac{dx}{dt}\).
   • Given \(\frac{dx}{dt} = 30 \, \mathrm{m/min}\), we substitute: 
     \(\frac{dy}{dt} = \frac{2}{3} \times 30 = 20 \, \mathrm{m/min}\).
5. Conclusion: The man's shadow is lengthening at a rate of \(20 \, \mathrm{m/min}\), which matches the correct answer option.