Question

A Norman window has the shape of a rectangle surmounted by a semicircle. (Thus the diameter of the semicircle is equal to the width of the rectangle.) If the perimeter of the window is 32 ft, find the value of x so that the greatest possible amount of light is admitted.

Answer 1

let x ≡ width of the rectangle, y ≡ height of the rectangle

A = xy + π/8 x² ................ area: rectangle + semicircle
x + 2y + π/2 x = 32 ......... perimeter: rectangle + semicircle

A = x(16 - x/2 - π/4 x) + π/8 x²
dA/dx = 16 - x/4 (4 + π)
16 = x/4 (4 + π) ............. set dA/dx = 0 to find stationary points
x = 64/(4 + π) .......... is a relative max since d²A/dx² < 0
y = 32/(4 + π)

Answer: 64/(4 + π) or approx. 8.96 ft (to 2 d.p.)

Answer 2

thank you again!(:

Answer 3

your welcome again

Answer 4

=)

Answer 5

a normal window has the shape of a rectangle surmounted by a semicircle. A window with perimeter 30 ft is to be constructed. find a function that models the area of the window.??? please help