calculus question

Question

A construction worker pulls a 5m plank up the side of a building under construction by means of a rope tied to the end of the plank. The opposite end of the plank is being dragged along the ground. If the worker is pulling at a rate of 15cm/s, how fast is the end of the plank sliding along the ground when it is 2m from the wall of the building? Express answer to 3 decimal places.

 

Help!?

Answer 1

It looks like you have a right triangle, where the 5m length is a constant hypotenuse, the base is x(t), and the height is h(t) = vt = (0.15 m/s)t.

 

h² + x² = 5²

 

(0.15)²t² + x² = 25

 

Take d/dt of both sides.

 

0.045t + 2xdx/dt = 0

 

dx/dt = = -0.045t / (2x)

 

When x = 2m,

 

h = √(25 - 2²) ≅ 4.58 m ⇒ 0.15t = 4.58 ⇒ t = 4.58/0.15 ≅ 30.5 s

 

dx/dt = -0.045(30.5) / [2(2)] m/s ≅ -0.343 m/s

 

The negative sign means the point is moving towards the wall.