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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

A boat is pulled into a dock by a rope attached to the bow of the boat and passing through a pulley on the dock that is 1 m higher than the bow of the boat. If the rope is pulled in at a rate or 1 m/s, how fast is the boat approaching the dock when it is 8 m from the dock?

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To determine

To find: The speed at which the boat is approaching the dock when it is 8 m from the dock.

Explanation

Given:

Since the rope is pulled at a rate of 1 m/s, that is dydt=1m/s.

Formula used:

Chain rule: dydx=dydududx

Calculation:

Let x be the distance between the boat and dock and y be the length of the rope which is attached to the bow of the boat and passing through a pulley on the dock as shown in the figure-1 given below.

Since x and y decreases with the time t, both x and y are functions of the time t.

Obtain the value of dxdt when the boat is 8 m from the dock.

Apply Pythagorean Theorem in Figure 1, y2=x2+1

If x=8, then the value of y is obtained as follows.

y=82+1=64+1=65m

Differentiate y with respect to the time t.

ddt[y2]=ddt[x2+1]2ydydt=2xdxdt

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