Question

A boat is pulled into a dock by a rope attached to the bow of the boat and passing through a pulley on the dock that is 1 m higher than the bow of the boat. If the rope is pulled in at a rate of 1m/s,how fast is the boat approaching the dock when it is 8 m from the dock?

Answer 1

draw the triangle
vertical distance
horizontal distance= sqrt(x^2-1)
hypotenuse= x

given dx/dt=1m/s
what is d(sqrt(x^2-1))/dt

d(hyp)/dt=1/2 * 1/sqrt(x^2-1) * 2xdx/dt

so when sqrt(x^2-1)=8, how fast then
1/2 (1/8)*2(sqrt(64+1)*1
= sqrt(65)/8 m/s

check my work.