Shrek saving Rapunzel (optimization)

[valve2021]

This involves some geometry as well which is what Im stuck on

Shrek is trying to save Rapunzel. She is in a 50 foot tall tower, with a 15 foot high wall standing 12 feet out from the tower's base,and an impenetrable moat is formed.

Thus, Shrek must use a ladder to lean against the wall of the moat. and it must bridge across enough to where he can free climb the tower. The thickness of the wall is negligible. Shrek's wall climbing abilities are exceptional and are not a factor.

What is the minimum ladder length Shrek can use?

 

[galactus]

There are various ways ro tackle this. Trig or proportions.

Draw a diagram of your ladder, wall , and tower.

Use Pythaogras to find the length of the ladder.

The base of the ladder from the tower is 12+x

The point where the ladder touches the tower above the ground is (15+y)

This forms a triangle with base 12+x and side 15+y. You need the length of the hypoteneuse, which is the ladder.

You can shed a variable by using similar triangles.

15/x=y/12------>y=180/x

So, the length of the ladder is:

\(\displaystyle \L\\L^{2}=(12+x)^{2}+(15+(\frac{180}{x}))^{2}\)

Differentiate and minimize.

I also used a trig method and got the same result, so you can try either.

There are different trig methods, but \(\displaystyle \L\\L=15csc({\theta})+12sec({\theta})\) will work.