Dustin S. answered • 11/16/16

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You need to picture a right triangle formed vertically by the dock, the base is the water and the hypotenuse is the rope.

The algebraic relationship between these three sides is the pythagorean theorem. a

^{2}+b^{2}=c^{2}.Now implicitly differentiate that relationship 2a(da/dt)+2b(db/dt)=2c(dc/dt).

We know the following:

a=1(the dock is a meter above the bow)

da/dt = 0 (that height is not changing)

b=7(the dock is 7 feet away from the boat)

db/dt = ? (that is the speed we are looking for

c = well...they didn't give it to use but we can find it using the pythagorean theorem because we know the dock is 1 meter and the boat is 7 m away so sqrt(1

^{2}+7^{2})=sqrt(50)=**5sqrt(2)**dc/dt = 1 (the rope is being pulled in at 1 m/s)

We can know plug into our differentiated equation

2(1)(0)+2(7)(db/dt)=2(5sqrt(2))(1)

and solve

db/dt = 5sqrt(2)/7