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.

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 1.4 m/s, how fast is the boat approaching the dock when it is 7 m from the dock?

Answer 1

construct a right triangle

with the rope length as the hypotenuse = c

base = 7 meters= b

height = altitude = 1 meter = a

c^2 = a^2 +b^2

take the derivative with respect to time

2cc' = 2aa' +2bb'

cc' = aa' + bb'

c' = 1.4 meters per second

a'=0

b=7, a=1, c= sqr(7^2+1^2) = sqr50

1.4sqr50 = 1(0) + 7b'

solve for b'

= (1.4/7)sqr50

=.2sqr50 meters per second

= about 9.9/7

= about 1.414 m/s