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 of 1 m/s, how fast is the boat approaching the dock when it is 7 m from the dock? (Round your answer to two decimal places.)

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Asked Oct 17, 2019
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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 of 1 m/s, how fast is the boat approaching the dock when it is 7 m from the dock? (Round your answer to two decimal places.)

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Expert Answer

Step 1

Refer to the question, from the above information one can note that the point where the rope touches the boat and pulley are the end points of the line segment that forms the hypotenuse of the right angled triangle. Also x(t) is the distance between ship and the dock and r(t) as length of the rope where as height of the dock is 1m . so sketch the diagram as,

Rope
pulley
1mis
1m
Ship
Dock
X
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Rope pulley 1mis 1m Ship Dock X

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Step 2

Now use Pythagorean theorem  as,

x21
+1=7
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x21 +1=7

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Step 3

Now perform implicit differentiation to find the ...

(*+1)= 4
dt
dx
dr
2x 2r
dt
dt
dк
rdr
dt
rdt
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(*+1)= 4 dt dx dr 2x 2r dt dt dк rdr dt rdt

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