Answer: 101.6 ft/min
Step-by-step explanation:
let x be the horizontal distance to the dock. And the rope is attached to the boat 10 feet below the pulley
Using pythagorean theorem
x^2=R^2-10^2
where R is rope length to the pulley.
Differentiates with respect to time t
2x dx/dt=2RdR/dt
Xdx/dt = RdR/dt ...... (1)
If the boat be approaching the dock when 125 ft of rope is out
This is R = 125ft
Using pythagorean theorem again
X^2 = R^2 - 10^2
X^2 = 125^2 - 10^2
X^2 = 15525
X = 124.6 ft
The rate the boat be approaching the dock = dx/dt
While
the rope is pulled through the pulley at a rate of 20 ft/min = dR/dt
Solve for dx/dt when R is 125, x= 124.6 and dR/dt= 20ft/min
Substitute all in equation 1
124.6 dx/dt = 125 × 20
dx/dt = 2500/124.6
dx/dt = 101.6 ft/min
