Aboat 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 1 m/s, how fast is the boat approaching the dock when it is 8 m from the dock?

boat is approaching the dock  at rate 1.007 m/s

Explanation:

Given data

dock that = 1 m

rate = 1 m/s

to find out

how fast is the boat approaching

solution

we apply here pythagoras theorem here

we know pulley is 1 m height

so horizontal distance is consider x and vertical is here 1 m

and hypotenuse is y

so  here we can say

x² +1² = y²     ..............1

if we differentiate w.r.t time

2x dx/dt = 2y dy/dy

so

x dx/dt = y dy/dt

we have given x = 8m and dy/dt is 1

so from equation 1

x² +1² = y²

y² = 8² +1²

y =

so now

x dx/dt = y dy/dt

8 dx/dt = (1)

dx/dt = (1) /8

dx/dt = 1.007 m/s

therefore we say boat is approaching the dock  at rate 1.007 m/s