Question

Docking Boat. A boat is

pulled to dock by a rope
attached to a pulley on the
dock. The vertical distance
between the pulley and
the boat is 3 metres. If the
rope is shortened at a rate
of 0.75 m/s, how fast is the
boat moving when the boat is
7.5 metres from the dock?

Answer 1

When the boat is 6 m from the dock, it is approaching the dock at the rate of √37/6 m/sec.

Explanation:

Let x = distance of the boat from the dock at time t

y = length of rope at time t

Draw a right triangle with horizontal leg x, vertical leg 1, and hypotenuse y.

We know that dy/dt = -1 and want to find dx/dt when x = 6.

By the Pythagorean Theorem, x2 + 12 = y2.

2x(dx/dt) = 2y(dy/dt)

dx/dt = y(dy/dt) / x When x = 6, y2 = 36 + 1 = 37

So, y = √37

dx/dt = -√37 / 6

When the boat is 6 m from the dock, it is approaching the dock at the rate of √37/6 m/sec.