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Relative Rate Problems

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30. A boat is pulled into a dock...
(a) Determine the speed of the boat when there is 13 feet of rope. What happens to the speed of the boat as it gets closer to the dock?
(b) Determine the speed of the rope when there is 13 feet of rope. What happens to the speed of the rope as the boat gets closer to the dock?

32. An airplane is flying at an altitude of 5 miles and passes directly over a radar antenna. ... The radar detects that the distance is changing at a rate of 240 mph. What is the speed of the plane?

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Please see attached file.

First, I made a diagram.....


12 ft


(a) From the diagram, you can see that we're dealing with a right angle, so let's set up an equation using the Pythagorean Theorem. Let r be the hypotenuse (the length of the rope) and let x be the bottom side (the distance that the boat is from the dock).

122 + x2 = r2

144 + x2 = r2

Now, let's differentiate the equation with respect to t (time):

0 + 2x(dx/dt) = 2r(dr/dt)

According to the question, the rope is getting shorter at a rate of 4 ft per second, so dr/dt = -4. We are also told that the rope is 13 feet long, so r = 13, ...

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