A Norman window is in the form of a rectangle below a semicircle. If the outside perimeter of the window is 10 ft, find the dimensions of the rectangle (length and width) so that the area of just the rectangle is maximized.
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jchu1314
ANSWERS: 1
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perimeter of half a circle = half pi times diameter = half pi times width of rectangle. Equation for perimeter = (half pi + 1) * width of rectangle + 2 * height of rectangle = 10 Quantity to maximize: width * height. Introduce two variables x = width*(half pi + 1) y = height*2 Now x + y = 10 and the quantity to maximize is x y multiplied by a constant - same as maximizing x y y = 10-x maximize 10 x - x^2 happens when 10 = 2 x x = 5 y = 5 width = 5/(half pi + 1) = 1.945 height = 2.5
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