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Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

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BuyFindarrow_forward

Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

A plane flying with a constant speed of 300 km/h passes over a around radar station at an altitude of 1 km and climbs at an angle of 30°. At what rate is the distance from the plane to the radar station increasing a minute later?

To determine

To find: The rate of change of the distance from the plane to the radar station.

Explanation

Formula used:

Chain rule: dydx=dydududx.

Calculation:

Let us assume that at any time t, A be the position of top of radar station on the ground and P be the position of the plane 1 minute later after crossing the radar and B be the position of the radar on the ground.

Let x be the distance between the top of the radar and plane and y be the distance between the ground point of the radar and the plane as shown in the Figure 1 given below.

Since x and y changes with the time t, x and y are the function of the time t.

Since the plane climbs at an angle of 30° from the horizontal, the total angle between the plane and radar is 120°.

Obtain dydt when t=1 min later.

From the cosine rule in the ΔAPB.

y2=12+x22(1)(xcos(120))=1+x22x(12)=1+x+x2

Differentiate y2=1+x+x2 with respect to the time t.

ddt(y2)=ddt(x2+x+1)2ydydt=(2x+1)dxdt             &#

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