this question is off a textbook, and I've been having a lot of trouble with it:
"A window consists of a rectangle surmounted by a semi-circle having its diameter the width of the rectangle. If the perimeter of the window is $t$ meters, find the greatest possible area of the window."
The final answer is:
$t^2 / 2(pi + 4)$
Could someone please explain how to do this step by step?
Thanks
Best Answer
We set things up, and leave the differentiation and conclusion to you.
Let us explore what happens if we let the radius be $x$. Then the curvy part of the semi-circle has perimeter $\pi x$. The base of the window has length $2x$. Since the whole perimeter is $t$, the two sides have combined length $t-\pi x-2x$. It follows that each side of the window has length $$\frac{t-\pi x-2x}{2}.$$
Next we find a formula for the area $A(x)$ of the window.
The semi-circle has area $\frac{\pi}{2}x^2$.
The rectangular part of the window has area $(2x)\frac{t-\pi x-2x}{2}=x(t-\pi x -2x)$. It follows that $$A(x)=\frac{\pi}{2}x^2 +x(t-\pi x -2x).$$ Now do the usual thing: find where $A'(x)=0$.
For completeness, we have to check whether the maximum is at an end-point, or alternately look at the sign of $A'(x)$ when $A'(x)\ne 0$.