A glass for a office building is being designed in the shape of a rectangle capped with a semi-circle. Determine the total cost of the glass required for this entire window to have a maximum area.

Question

The perimeter of the window is not to exceed 30 m. The price of the stained glass is $9.75/m^2.

this is for my quadratic relations homework.

Answer 1

Let r=radius of the semicircle, so the width of the glass is 2r. Let length of the rectangular part be x. The perimeter is πr+2x+2r≤30m.

So 2x≤30-πr-2r=30-r(π+2).

Area of the glass is A=πr²/2+2rx=πr²/2+r(30-πr-2r)=if the perimeter is 30m.

The derivative dA/dr=0=πr+30-2πr-4r=30-πr-4r at maximum or minimum.

Therefore, r=30/(π+4)=4.20m, and 2x=30-30(π+2)/(π+4)=60/(π+4)=8.40m.

Note that this means we have a square glass with a semicircular top, so x=r.

A=πr²/2+2r²=r²(π/2+2) m²

Area of the glass is 63.01m² approx, costing $614.36 approx.