Question

How fast is the boat approaching the dock when...? (See entire problem below)

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

Answer 1

`approx - 1.01 \ ms^(-1)`

Explanation:

Use Pythagoreas Theorem in terms of the distances `x,y,1`, as displayed in the drawing:

`y^2 = x^2 + 1^2`

Differentiate both sides wrt t:

`2 y dot y =2 x dot x`

We want `dot x:`

`dot x = y/x dot y = (sqrt(x^2 + 1))/x \ dot y`

`= (sqrt (7^2 + 1))/7 cdot (-1) approx - 1.01 \ ms^(-1)`

The sign is negative because the distance from the pier is decreasing.