Question

A handyman knows from experience that his 29-foot ladder rests in its most stable position when the distance of its base from a wall is 1 foot farther than the height it reaches up the wall. How far up a wall does this ladder reach?

Answer 1

This 29-foot ladder forms a triangle, with the height of the top of the ladder and the distance of the base of the ladder from the wall forming the shorter sides.  According to the Pythagorean Theorem, (base)^2 + (height)^2 = (29 ft)^2.  

But here we're told that (base) = (height) + 1 ft.

Substituting (height) + 1 ft for (base), we get:

(height)^2 + [(height) + (1 ft)]^2 = (29 ft)^2

Expanding  [(height) + (1 ft)]^2, we get:

(height)^2 + (height)^2 + 2(height) + (1 ft)^2 = (29 ft)^2, or

2(height)^2 + 2(height) + 1 ft^2 = 841 ft^2.  This equation has the form

2h^2 + 2h - 1 = 841, or 2h^2 + 2h - 842 = 0.  This is a quadratic equation with a = 2, b = 2 and c = -842.

The discriminant is b^2-4ac, or 4-4(2)(-842) = 4+6736, or 6740, and the sqrt of the discriminant is 82.1 ft.

Thus,

                 -2 plus or minus 82.1 ft

(height) = -------------------------------------

                               2(2)

(height) =  [(height) + (1 ft)]^2 = 20.0 ft, or (height) = (-2 - 82.1)/4 = negative #

We must discard the negative root because it does not make sense in this situation.  Thus, the top of the ladder reaches the building 20 ft. above the ground.