Calculus
posted by Anonymous .
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 30ft, find the dimensions of the window so that the greatest possible amount of light is admitted.
What I have so far:
C = 1/2*pi*d, where d=x, or the width of the rectangle
P(rectangle) = 2y + x
So, the total perimeter:
P = 2y + x + 1/2*pi*x = 30ft
I'm not sure what to do next. How do I make this expression in terms of x?
Thank you

area = A = pi x^2/8 + x y
so
y = [A  pi x^2/8]/x
so
30 = 2 [A  pi x^2/8]/x + x + .5 pi x
30 x = 2 A  pi x^2/4 + x^2 + .5 pi x^2
(1+pi/4)x^2  30 x = 2 A
parabola, find vertex for max
1.785 x^2  30 x = 2 A
x^2  42 x = 1.12 A
complete square
x^2 16.8 x + 70.6 = 1.12 A + 70.6
(x8.4)^2= 1.12 A + 70.6
= 1.12 (A 63.3)
so x = 8.4 at vertex
and A = 63.3
so find y from
y = [A  pi x^2/8]/x
= [ 63.3  pi 70.6/8] /8.4
=4.23
Check my arithmetic !
=
Respond to this Question
Similar Questions

Math
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 24 ft, find the value of x so that the greatest … 
calculus
A Norman window has the shape of a rectangle surmounted by a semicircle. If the perimeter of the window is 30ft, find the dimensions of the window so that the greatest possible amount of light is admitted. I keep screwing up in creating … 
Calculus
A Norman window has the shape of a semicircle atop a rectangle so that the diameter of the semicircle is equal to the width of the rectangle. What is the area of the largest possible Norman window with a perimeter of 40 feet? 
Calculus
A Norman window has the shape of a semicircle atop a rectangle so that the diameter of the semicircle is equal to the width of the rectangle. What is the area of the largest possible Norman window with a perimeter of 30 feet? 
Calculus
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 25 ft, find the dimensions of the window so that … 
Math
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 30 ft, find the dimensions of the window so that … 
Calculus
A Norman window has the shape of a semicircle atop a rectangle so that the diameter of the semicircle is equal to the width of the rectangle. What is the area of the largest possible Norman window with a perimeter of 33 feet? 
Calculus
A Norman window has the shape of a semicircle atop a rectangle so that the diameter of the semicircle is equal to the width of the rectangle. What is the area of the largest possible Norman window with a perimeter of 49 feet? 
calculus
A Norman window has the shape of a semicircle atop a rectangle so that the diameter of the semicircle is equal to the width of the rectangle. What is the area of the largest possible Norman window with a perimeter of 31 feet 
Calculus  Optimization
A Norman window has the shape of a semicircle atop a rectangle so that the diameter of the semicircle is equal to the width of the rectangle. What is the area of the largest possible Norman window with a perimeter of 38 feet?