The winch is pulling in rope at a rate of 1.5384 ft/sec.

[nyr]

Theres two parts to the question. Each part also has two parts to it. I am pretty sure my work i did so far is correct( double check please) but I am not sure about the bolded parts

A boat is pulled into a dock by means of a winch 12 feet above the deck of the boat.?


a) The winch pulls in rope at a rate of 4 feet per second. Determine the speed of the boat when there is 13 feet of rope out. What happens to the speed of the boat as it gets closer to the dock?


b) Suppose the boat is moving at a constant rate of 4 feet per second. Determine the speed at which the winch pulls in rope when there is a total of 13 feet of rope out. What happens to the speed at which the winch pulls in rope as the boat gets closer to the dock?
______________________________________________________


a) The boat is 5 ft from the dock. (5-12-13 rt triangle)
Let a = distance from dock, b = height of dock, and c = length of rope.

a^2 + b^2 = c^2
2a da/dt + 2b db/dt = 2c dc/dt
a da/dt + b db/ddt = c dc/dt
5 da/dt + 12(0) = 13 • -4
5 da/dt = -52
da/dt = -10.4
The boat is going -10.4 ft/sec.


b) a da/dt + b db/dt = c dc/dt
5 • 4 + 12(0) = 13 dc/dt
20 = 13 dc/dt
1.5384... = dc/dt

 

[lippyka]

The winch is pulling in rope at a rate of 1.5384 ft/sec.As the boat gets closer to the dock, the speed at which the winch pulls in rope will decrease.