# Thread: Norman window maximum problem

1. ## Norman window maximum problem

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.

2. Yes, you may substitute $15-r-\frac{\pi r}{2}$ for $y$, as an increase in $r$ implies a corresponding decrease in $y$, and the area function $A$ may therefore be written in terms of $r$ alone.

3. Hi,
I meant if I can write y = 15-r-(pi*r)/2 so that there is only one term of r in y not in the area formula.

4. Yes, you may factor out $r$:

$y=15-r\left(1+\frac{\pi}{2}\right).$