Need help with Norman Windows Problem (pre-calculus)?

Question

A Norman Windows has the shape of a rectangle surmounted by a semicircle of the diameter equal to the width of the rectangle. If the perimeter of the windows is 20ft, what dimenstions will admit the most light (maximize the area)?

Hint: Circumference of a circle = 2*pi*r ; area of a circle = pi*(r^2), where r is the radius of the circle.

Answer 1

Let x be the radius of the semicircle, then the rectangle has base 2r and height h. The total perimeter of the window is

pi*r + 2r + 2h = 20.

So h = (20 - pi*r - 2r) / 2

The area is (pi*r^2)/2 + 2rh. Replace h with the above so that the area is a function of r. Differentiate to find the maximum.

Answer 2

This would be a fun problem to solve, but difficult to do without drawing and writing out on paper, so I won't even attempt to do it via typing. However, here are some ideas to get you started on how to solve this:

draw the shape (rectangle with a semi-circle on top) and label the bottom (width) as x and the side as y

so now you know the total perimeter of the window is x + 2y + 2pi * x, so x + 2pi x + 2y = 20

The area would be x*y + pi/2 x^2

You want to find values for x and y that maximize area. This should be enough to get you on the right track :)

Best of luck!

Answer 3

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