trigonometry - Converting a word problem into an equation? Trignometry and calculus

The problem is question 3 of what I am about to download into this question. I drew a diagram of what the problem actually is, my professor has verified it's correct. I don't want an exact answer to this question, just a way to convert question 3 into an equation so that I can use programming to solve. I have software I have created that can solve for zeros of functions and minimums of functions numerically.

I am thinking that this is a problem of looking for some type of maximum because we want the biggest ladder that can fit through. Is this correct?

It's probably a nooby question, but my trignometry is pretty rusty. I have a feeling it will be blatantly obvious

Answer

Here's how I think of it. You can break it into two triangles, one with base $7$, hypotenuse $L_1$ and angle $\theta \neq 0$ and a second triangle of height $9$, hypotenuse $L_2$ and identical angle $\theta$. (Depending on where you put theta) you get the equations

$$\cos(\theta)=\frac{7}{L_1}$$ and $$\sin(\theta)=\frac{9}{L_2}$$ use these two equations to get $L = L_1+L_2$. You can minimize $L$ and get your answer.