Local extrema

Imagine you have a house with a backyard in the shape of a semicircle, but you're not really using it for anything. It gets plenty sunlight, so you decide to install a solar panel. However, your local manufacturer only makes rectangular solar panels. What dimensions must your panel have in order to maximize its area knowing that the radius of the semicircle is $30\sqrt2$ meters?

Remember:

• A semicircle can be described as: $y = \sqrt{r^2 - x^2}$
• $A_{rectangle} = base \cdot height$

Tip: If we set the origin of coordinates at the center of the circle, we can see that the base of our rectangle is: $base=2x$ since the semicircle is symmetric about the y-axis, and its height is: $height = y$