A Norman Window has the perimeter of 30ft, find the dimentions of the window so that the greatest possible amount of light is admitted.

I did:

Perimeter of half circle = pi*r

Perimeter of the rectangle (3sides) 2r + 2y

Perimeter of the window

30 = pi*r + 2y + 2r

Solve for y

y = 15 - r - (pi*r)/2

I understand out of the text that I need the max area of the window:

Area half circle = (pi*r^2)/2

Area of rectangle = 2r * y

Area of the window A= (pi*r^2)/2 + 2r*y

Now I plug in y

A= (pi*r^2)/2 + 2r* [15 - r - (pi*r)/2]

My question:

Can I write y = 15 - r - (pi*r)/2 so that I can combine the two terms with r in one term?

Please, including the way you did that.

And what is the derivative then? A'(r)?

Please, do not give me any links of similar problems, I looked at them but this did not help me with my problem.

Thanks.