Calculus Problem

Question

A construction worker pulls a five-meter plank up the side of a building under construction by means of a rope tied to one end of the plank. Assume the opposite end of the plank follows a perpendicular path to the wall of the building and the worker pulls the rope at a rate of 0.15 meters per second. How fast is the end of the plank sliding along the ground when it is 2.5 meters from the wall of the building?

Answer 1

Hi Savannah W.,

This is a medium-level-difficulty problem, because you could apply both your geometry and your calculus to it, or not. So, as an algebra-type approach:

1) First, sketch your problem at the conditions stated. You will see a right triangle described by the ground, the wall, and the plank. You can assign lengths immediately to two of those legs, and then (Pythagoras, remember?) calculate the third leg.

2) Now write the equation linking small changes in the vertical and horizontal legs, letting "x" represent the vertical change in the chained end, and "y" represent the horizontal change in the free end, of the plank. This is still a Pythagorean formula!

3) Now you can differentiate your equation, and relate d(x) to d(y).

But that's the hard way!

Now do it as geometry:

1) Let the top-most angle of the triangle be .theta.

2) Then the vertical change is d(cos(.theta.))/d(.theta.), and the horizontal change is d(sin(.theta.))/d(.theta.). These equations follow naturally from the definitions of the trig functions, and the given triangle. Do the differentials (it's cool that the trig functions differentiate into each other, given a +/- reversal, isn't it!)

3) So ratio these two, and express for .theta. = .pi./6 (i.e., 30 .degrees.). Plug into the given conditions, and there you are.

Now for the extender question! If the ground were frictionless with the free end of the plank, but the wall was not frictionless with the chained end of the plank, at what angle of inclination would the plank have maximal friction with the wall? You can see that it must be somewhere "in the problem", since there would be no force horizontally when the plank is horizontal=level, and none when the plank is vertical. But there must be some between these extremes, b/c the plank is leaning against the wall (the lifting force only counteracts the vertical component of friction (which is horizontal in frame of reference of the wall). Right?

Cheers, -- Mr. d.