Question

How do you find the dimensions of the rectangle with largest area that can be inscribed in a semicircle of radius r ?

Answer 1

Let length of the side be x, Then the length of the other side is 2r2−x2−−−−−−√, as shown in the image.

Rectangle inscribed in a semi circle

Then the area function is

A(x)=2xr2−x2−−−−−−√

A′(x)=2r2−x2−−−−−−√−4xr2−x2−−−−−−√=2r2−x2−−−−−−√(r2−2x−x2)

setting A′(x)=0,

⟹x2+2x−r2=0

Solving, I obtained:

x=−1±1+r2−−−−−√

That however is not the correct answer, I cannot see where I've gone wrong? Can someone point out any errors and guide me the correct direction. I have a feeling that I have erred in the differentiation.

Also how do I show that area obtained is a maximum, because the double derivative test here is long and tedious.

Thanks!