Calculus problem (begining calculus)?

Question

A Norman window has the shape of a rectangle surmounted by a semicircle. If the perimeter of the window is 39 ft, find the value of x so that the greatest possible amount of light is admitted. (Give your answer correct to two decimal places.)

Answer 1

If the circle has radius r, the rectangle has base length 2r. Let h = rectangle height.

The perimeter is p = 2*r + 2*h +π*r (just add the lengths of the line segments)

The area is A = 2*r*h +0.5*π*r^2 (add the rectangle area to the half circle area)

Find A in terms of the perimeter and r by eliminating h:

From the first equation

2h = p - (2 + π)*r

put this in for 2h in the area equation

A(r) = r*[p - (2 + π)*r] + 0.5*π*r^2

multiply out and combine terms:

A(r) = p*r - (2 + π)*r^2 + 0.5*π*r^2

A(r) = p*r - (2 + 0.5*π)*r^2

Now take the derivative of A with resp to r:

dA(r)/dr = p - 2*(2 + 0.5*π)*r

dA(r)/dr = p - (4 + π)*r

the maximum area occurs when dA(r)/dr = 0

p - (4+ π)*r = 0

r = p / (4 + π)

From the perimeter equation

2*h = p - (2 + π)*r

h = [p - (2 + π)*r ]/2

I don't know whether x was r or h, but it is one or the other.

Answer 2

You didn't specify which dimension x refers to, so I solved the problem in terms of the radius of the semi-circle. If the semi-circle has radius r, then the rectangle has dimensions 2r x height. Hopefully the rest makes sense to you.

height = .5*(39 - pi*r - 2 r)

Area = height*2*r + .5* pi*r^2 = (39-pi*r - 2 r)*r + .5*pi*r^2 =
39r - pi*r^2 - 2 r^2 + .5 pi*r^2 = -(2 + .5 pi)r^2 + 39 r

Take the derivative and find the zeros,

A'(r) = -(4 + pi)r + 39 = 0 =>

r = 39/(4+pi)