   Chapter 3.9, Problem 45E ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

#### Solutions

Chapter
Section ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

# A plane flies horizontally at an altitude of 5 km and passes directly over a tracking telescope on the ground. When the angle of elevation is π/3, this angle is decreasing at a rate of π/6 rad/min. How fast is the plane traveling at that time?

To determine

To find: The speed of the plane.

Explanation

Given:

Since the plane flies horizontally at an altitude of 5 km and passes directly over a tracking telescope over the ground.

The rate at which the angle of elevation is decreases π6rad/min.

Formula used:

(1). Chain rule: dydx=dydududx

Calculation:

Let at any time t, P be the position of the plane and A be the position of the telescope on the ground and B be the point on the ground just below the plane. And x be the distance between the telescope and point on the ground just below the plane and θ be the angle of elevation between the telescope and plane as shown in the Figure 1 given below.

Since the plane is flying horizontally.

Therefore, the horizontal distance x and the angle of elevation changes with the time t.

Since the angle of elevation decreasing at a rate of π6rad/min.

In the ΔABP

cotθ=ABPB=x5x=5cotθ

Differentiate x with respect to the time t

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