Question

Find the dimensions of largest rectangle w/ maximum area inscribed in a semicircle with radius r.
What equations am I gonna use in here? Help!

Answer 1

The equation of the semicircle is
\[x^2+y^2=r^2\]
The area will be
\[A=2xy\]
So
\[y=\sqrt{r^2-x^2}\]
and
\[A(x)=2x\sqrt{r^2-x^2}\]

Answer 2

It's now time to use differential calculus so that you can maximize A. What is dA/dx?

Answer 3

\[\frac{dA}{dx}=2\sqrt{r^2-x^2}-\frac{2x^2}{\sqrt{r^2-x^2}}=\frac{2r^2-4x^2}{\sqrt{r^2-x^2}}\]
\[\frac{2r^2-4x^2}{\sqrt{r^2-x^2}}=0\]
\[x=\frac{r}{\sqrt{2}}\]