The upper end of an 8 m ladder rests against a vertical wall with the lower end on the horizontal ground. The lower end of the ladder slips along the ground away from the wall at a rate of 6 m/s. Find the rate at which the upper end of the ladder is moving the instant the ladder is 4 m from the wall.
I'm defining the top point of the ladder on the vertical wall to be height
H, and the distance from the wall to be
L.
We also know the ladder will be constantly 8m long, and that it is moving away from the wall at a rate of 6m/s, therefore
The question is asking us to find the rate at which the height of the ladder is changing, or
Using the chain rule, we can get that
Since we can sorta pretend these are fractions, something will equal
, which via cancelling will give us what we want.
All we need to do know is find an expression for
, so it will involve H and L. First thing that comes to mind is Pythagoras's Theorem!
Taking the positive root as a length cannot be negative.
Now all we need to do is differentiate to get our
Going back to
, we substitute in our derivatives we found to get
The question asks us to find the rate when the ladder is 4m from the wall, so L = 4.
So when the ladder is 4m away from the wall, the height of the ladder is decreasing at