Window dimensions calc question?
A Norman window has the shape of a rectangle surmounted by a semicircle. Find the dimensions of a Norman window of perimeter 21 ft that will admit the greatest possible amount of light.
A Norman window has the shape of a rectangle surmounted by a semicircle. Find the dimensions of a Norman window of perimeter 21 ft that will admit the greatest possible amount of light.
Quevvy
I'm not sure how in-depth I should go for this.
Assume the radius of the semicircle is r (diameter is d), and the other dimension of the rectangle is w. Thus, the perimeter is 2*w + d + πd/2 = 21. From the perimeter, we can solve for w.
w = (21 - d - πd/2)/2
The area is d*w + π*d^2/8, or d*(21 - d - πd/2)/2 + π*d^2/8.
This can be simplified to d^2*(π/8 - 1 - π/2) + 21d = 21d - (1+3π/8)d^2.
To maximize this, take the derivative and set to zero, dA/dd = 21 - (2+3π/4)d = 0.
Thus d = 21/(2+3π/4) ≈ 4.821.
And w ≈ 4.303 (exact solution is messy, but easily found from earlier expression for w.)