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Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550

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Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550
Textbook Problem

The angle of elevation of the sun is decreasing at a rate of 0.25 rad/h. How fast is the shadow cast by a 400-ft-tall building increasing when the angle of elevation of the sun is π/6?

To determine

To find: The increasing rate of length of the shadow.

Explanation

Calculation:

Let θ is the angle of the elevation of the sun and x is the length of the shadow.

The angle of elevation of the sun is decreasing at a rate of 0.25 rad/h.

That is, dθdt=0.25 rad/h (negative sign denote the decreasing rate).

The height of the tall building is 400 ft.

From Figure 1,

cotθ=ACBCcotθ=x400x=400cotθ

Differentiate with respect to t,

dxdt=ddt(400cotθ)dxdt=400(csc2θdθdt)

Substitute dθdt=0

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