What dimensions will result in the rectangular portion of the window having the largest possible area

Question

A norman window is in the shape of a semi-circle atop a rectangle, as shown. The distance around the outside of the window is 40 feet. What dimensions will result in the rectangular portion of the window having the largest possible area

Answer 1

Let x = width of the rectangle and y = height of the rectangle.

x + 2y + (1/2)πx = 40

So, 2y = 40 - x - (1/2)πx.

y = 20 - (1/2)x - (1/4)πx

Area of rectangle = A = xy = 20x - (1/2)x² - (1/4)πx²

A' = 20 - x - (1/2)πx = 20 - [(2 + π) / 2]x = 0

x = 40 / (2 + π) and y = 20 - 20 / (2 + π) - 10π / (2 + π)

Answer 2

Most of the solution is in the labeling!

Label the width of the window "w". w

Length of a semicircle is wπ

The height of the widow's rectangle is (40 - w - wπ) / 2

Area of rectangle is w [ (40 - w - wπ) / 2 ]

Simplify, combine like terms, and differentiate.

Answer 3

Perimeter = 40 feet

pid+ d =2h =40

h= 20-((pi+1)/2)d

find dimensions that have the rectangle with largest area

height of the rectangle = about 12.22

base = about 7.78

rectangle's max area = about 95.1