Hi,
This is a very challenging problem. I've been working on this for a while. (it was a word document at first).
Thanks. Any help is greatly appreciated
The first thing to do is give names to some of the unlabeled distances in the picture. Lets call the height of the short ladder $\displaystyle h_1$ and the height of the long one $\displaystyle h_2$. We can also divide $\displaystyle d$ into two parts where it intersects the dotted red line; call the one on the left $\displaystyle d_1$ and the one on the right $\displaystyle d_2$.
Now we have two pairs of similar right triangles; each ladder forms a triangle with the building and with the dotted red line. We can set up some proportions for these triangles:
$\displaystyle \frac {d} {h_1} = \frac {d_2} {c} $
$\displaystyle \frac {d} {h_2} = \frac {d_1} {c} $
By multiplying both sides of each equation by $\displaystyle c$ and seeing that $\displaystyle d_1 + d_2 = d$ we have
$\displaystyle \frac {d c} h_1 + \frac {d c} h_2 = d$
which after dividing both sides by $\displaystyle d c$ becomes
$\displaystyle \frac 1 h_1 + \frac 1 h_2 = \frac 1 c$
Now all we have to do is use the Pythagorean Theorum to get $\displaystyle h_1 = \sqrt {20^2 - d^2}$ and a similar expression for $\displaystyle h_2$ and substitute.
The easiest way to do part b is to plug in 12 for $\displaystyle d$, then simplify the left side before taking the reciprocal of each side.
Part c is basically guess and check; use a calculator or computer program to plug lots of two-decimal-place values into that expression until you can change the sign by changing the plugged in value by only .01.