posted by Alex .
A Norman window consists of a rectangle capped by a semi-circular regional. The perimeter of some particular Norman window must be 30 meters. The radius of the semi-circular region is "x" meters. The height of the rectangle is "h" meters. Find the values pf "x" and "h" such that this Norman window has a maximum area. That is, the Norman window will permit the most light to shine through.
If the radius is x, then the base of the rectangle is 2x
let the height of the rectangle be h
Perimeter = 30 = (1/2)(2πx) + 2h + 2x
30 = πx + 2h + 2x
h = (30 - πx - 2x)/2
Area = A = 2xh + (1/2)πx^2
= 2x(30 - πx - 2x)/2 + (1/2)πx^2
I will let you finish it ...
simplify a bit, then differentiate,
set the derivative equal to zero and solve for x