Architectural Designs

Help me with my math homework... 10pts to best answer!?

A stained glass window for the entrance of Matt's new office building is to be designed by Stacey. Stacey decides to design the window in the shape of a rectangle capped with a semi circle. The perimeter of the window is to be 30m. The price of the stained glass is $9.75/m squared. Determine the total cost of the glass required for this window to have a maximum area?

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1. Let the height be h, the width be w. The diameter of the semi-circle is equal to w. Keep exact values until the end to avoid compounding roundoff errors. The area of a circle with diameter w is (1/4)π*w², and it's perimeter is π*w. The area is the area of the rectangle plus half of a circle: Area = h*w + (1/8)*π*w² The perimeter is: Perimeter = 2*h + w + (1/2)*π*w = 30 Solve for h from the perimeter, then simplify: h = 15 - [(1/2) + (π/4)]w and plug this into the formula for area, the simplify: Area = 15w - [(1/2) + (π/8)]w² Now take the derivative of the area and set it to zero: d(Area)/dw = 15 - [1 + (π/4)]w = 0 w = 60/(4 + π) ≈ 8.4 m So h = 30/(4 + π) ≈ 4.2 m And the area = 450/(4 + π) ≈ 63.01115 m² So the cost is: 450/(4 + π) * (39/4) ≈ $614.36
2. Alright, so this is using applied maximum and minimum if you're looking for the least cost for the required perimeter. Let's assign the window some variables ____ |.......| |___| x ...y Now The total area would be xy + (x/2)^2*pi The total perimeter would be x + 2y + (x * pi)/2 = 30 Using the perimeter equation, let's isolate one variable. 2y = ( 30 - x - (x * pi)/2 ) y = ( 30 - x - (x * pi)/2 ) / 2 Substitute into the Area equation x( 30 - x - (x * pi)/2 ) + (x/2)^2*pi 30x - x^2 - (x^2 * pi)/2 + (x^2/4)*pi = A Now find the derivative of A, as that will show when the instantaneous slope = 0, thus indicating a maximum/minimum. A' = 30 - 2x - x*pi + (x*pi)/2 0 = 30 - 2x - x*pi + (x*pi)/2 -30 = - 2x - x*pi + (x*pi)/2 -30 = -x(2 + pi - pi/2) -30 / 3.57 = -x x = 8.4 (appr) then y = 4.2 (appr) Then, your total area would be 49.13m^2, thus your cost would be 479 dollars. Whew, thanks for the question. Took me a heck of a lot of time.