Center your semicircle at the origin and inscribe your rectangle.
Draw a line from the origin up to the corner of the rectangle. You can
extend the y-axis upward and then break your rectangle into 2 halves.
This gives the area of the rectangle:
A=2xy...[1]
The equation of the semicircle is \(\displaystyle \L\\y=\sqrt{16-x^{2}}\)..[2]
Sub [2] into [1]:
\(\displaystyle \L\\A(x)=2x(\sqrt{16-x^{2}})\)
Differentiate(think product rule), set to 0 and solve for x.
Your dimensions will follow.
There's a small trick that may be helpful to check yourself.
The largest rectangle that can be inscribed in a semicircle of radius R has
an area of \(\displaystyle \frac{2}{\pi}\) that of the semicircle.
In other words, find the area of your semicircle, multiply by \(\displaystyle \frac{2}{\pi}\) and that'll give you the area of the largest rectangle that can be inscribed.
The area of your semicircle is \(\displaystyle \frac{\pi\cdot{4^{2}}}{2}=8{\pi}\)
\(\displaystyle 8{\pi}\cdot\frac{2}{\pi}=16\)
Go through the calculations and see if you get that.
Notice, that's also equal to the square of the radius.
It's a shame more people don't see mathematics for the beautiful thing it
is instead of having hangups about hating it or being afraid.
Relax, it'll be OK. And stop procrastinating.
