SOLUTION: A window is being built and the bottom is a rectangle and the top is a semicircle. If there is 12 m of framing materials what must the dimension of the window be to let in the most

Question 1005984: A window is being built and the bottom is a rectangle and the top is a semicircle. If there is 12 m of framing materials what must the dimension of the window be to let in the most light?
Note: This is a calculus optimization problem. I need to solve it algebraically, not with calculus (ex: first derivative test...). Thanks. I appreciate your help.
Answer by josgarithmetic(39617) (Show Source): You can put this solution on YOUR website!
Imagine that amount of light is according to window area.
Bottom of rectangle is x, height of rectangle is y, radius of the semicircle is x/2.

Length of the framing materials is
and the area of the window is .

That should get you started. Either solve the framing equation for x in terms of y and substitute into the A function; or solve the framing equation for y in terms of x and substitute into the A function; and then simplify the function. Is it quadratic, or does it have a quadratic numerator?

--SUMMARY OF PART OF THE METHOD, NOT TO COMPLETION:
Solve the framing equation for y,
;
Substitute into the area function A and simplify, very detailed work steps, to get .
This is parabola with vertex at a maximum, and it occurs exactly in the middle of the two zeros of A, which is why A is shown in its factored form, so you can more easily identify the zeros.
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