A rope passing through a capstan on a dock is attached to a boat offshore.

Question

A rope passing through a capstan on a dock is attached to a boat offshore. The rope is pulled in at a constant rate of 2 ft/s, and the capstan is 5 ft vertically above the water. How fast is the boat traveling when it is 10 ft from the​ dock?

Let x be the horizontal distance from the boat to the dock and z be the length of the rope. Write an equation relating x and z.

Answer 1

z² = x² + y²

z = √x² + y²

We know...

1. y = 5ft

2. dz = 2 ft/s

3. x = 10ft

Derive: z² = x² + y²

2z*dz = 2x*dx (y is a constant so it equals 0)

Rearrange: dx = (z*dz)/x

Plug variables: dx = (√x² + y² * dz)/x --> dx = (√10² + 5² * 2)/10

Solve: (2√125)/10 or √125/5