Related Rates Question

Question

A boat is pulled by means of a winch on a dock 8 ft above the deck of the boat. The winch pulls in the rope at the rate of 5 ft/s. Determine the speed of the boat when there is 10 ft of rope out.

Answer 1

PROBLEM

Related Rates Question

A boat is pulled by means of a winch on a dock 8 ft above the deck of the boat. The winch pulls in the rope at the rate of 5 ft/s. Determine the speed of the boat when there is 10 ft of rope out.

SOLUTION

Winch O

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| \

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y = 8 ft | \ Rope = c = 10 ft

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Dock | \

-------------| x = 6 ft \ ___________ Boat

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Using Pythagorean Theorem, to find the length of the rope from the winch to the boat:

x² + y² = c²

Taking the derivative:

d/dt [x² + y² = c²]

2x(dx/dt) + 2y(dy/dt) = 2c(dc/dt)

When there is 10 ft of rope out, we know:

dc/dt = -5 ft/s (GIVEN)

c = 10 ft (GIVEN)

y = 8 ft (GIVEN)

x = 6 ft (Found using 3-4-5 Triangle or Pythagorean Theorem)

dy/dt = 0 (Height of winch does not change)

Substituting the values into 2x(dx/dt) + 2y(dy/dt) = 2c(dc/dt)

2•6(dx/dt) + 2•8(0) = 2•10(-5)

12(dx/dt) + 2•8(0) = -100

dx/dt = -100/12 = -25/3 ft/s

Therefore, the boat is moving a speed of 8.33 ft/s toward the dock.