I know that If I were to make a loose coordinate plane graph than the radius and (X,Y) of the rectangle would have to mixed into make an equation out of the whole thing. What exactly that equation is beyond me though.
Can someone help me make an equation out of this? I asked my professor and he said that the key to find the local Min-Max.
I'm not given any specific numbers either. Any ideas?
Best Answer
Let the equation of semicircle be given by $x^2+y^2=r^2$ (Consider the upper half part)
Now, let the length of side of rectangle be $2a$ and width $b$ vertexes of rectangle will be $$(a,0)~ ;~ (a,b) ~;~ (-a,0) ~; ~(-a,b)$$
Also, $a^2+b^2=r^2$ (The point $(a,b)$ lies on the circle) , Put in (1)
$A=2ab=2a\Big(\sqrt{r^2-a^2}\Big)=2\sqrt{r^2a^2-a^4}=2\sqrt{\dfrac{r^4}{4}-\Bigg(a^2-\dfrac{r^2}{2}\Bigg)^2} \le 2\sqrt{\dfrac{r^4}{4}}$
$$\boxed{A \le {r^2} }$$