A boat is being pulled to a dock by a rope attached to a winch that is located on the dock, 8 feet above the water. If the winch is pulling the rope in at a rate of -1.5 ft./sec, what is the rate of the boat when it is 6 feet away from the dock? rope 8 ft. ? ft./sec ​

3.1 ft/sec

Explanation:

we are talking about a right-angled triangle.

the Hypotenuse c is the line (rope) connecting the boat and the winch.

the side a is the height of the winch (8 ft straight up from the water level).

the side b is the distance of the boat to the dock on the surface of the water. 6ft.

c² = 8² + 6² = 100

c = 10 ft

when pulling in 1.5 ft of the rope, I create a new triangle, where side a is still the same, the new side c is 1.5 ft shorter than the original one, and we need to calculate the new side b based on that.

the difference of original b and new b is then the distance per second (= the rate) of the boat being pulled at that point.

so,

(10-1.5)² = 8² + b²

8.5² = 64 + b²

72.25 = 64 + b²

8.25 = b²

b = 2.9 ft

original b - new b = 6 - 2.9 = 3.1 ft

=> the boat's rate is 3.1ft/sec