Question 56081: 4. A boat is pulled by a winch on a dock, and the winch is 12 feet
> above the deck of the boat. The winch pulls the rope at a rate of 4
> feet per second.
>
> a. Find the speed of the boat when the rope is at a length
> of 15 feet.
> b. What happens to the speed of the boat as it gets closer
> and closer to the dock?
>
Answer by stanbon(75887)   (Show Source):
You can put this solution on YOUR website! A boat is pulled by a winch on a dock, and the winch is 12 feet
> above the deck of the boat. The winch pulls the rope at a rate of 4
> feet per second.
>
> a. Find the speed of the boat when the rope is at a length
> of 15 feet.
> b. What happens to the speed of the boat as it gets closer
> and closer to the dock?
---------------
Draw the picture; it is a right triangle with rope as the hypotenuse,
boat as the base and 12 as the altitude.
EQUATION:
r^2=b^2+12^2
Take the derivative wrt time to get:
2r dr/dt = 2b db/dt
r (dr/dt) = b (db/dt)
----------------
You are told dr/dt = 4 ft/sec
So, 4r= b(db/dt)
And db/dt = 4r/b
--------------
Question a:
If the rope is 15 you find b=9 using Pythagoras: 15^2=12^2+b^2
Then db/dt = 4(15)/9 = 20/3 ft. per sec.
That is the speed of the boat when the rope is 15 ft.
---------------
Question b:
As the boat gets close to the dock the rope gets shorter
but not less that 12. Then b gets smaller because it is
the base of the rectangle which has a smaller hypotenuse.
The smallest b can be is zero. As b approaches zero
the fraction r/b goes to infinity. So the boat goes faster
and faster as the rope gets shorter.
Cheers,
Stan H.
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