The area of a (full) circle is (with being circle radius). I think there's a factor missing in your formula for the area.
This problem is giving me some trouble. It's irritating because it seems so easy.
The question is
"The figures are made of rectangles and semicircles.
a.) Find a formula for the area
b.) Find a formula for the perimeter
c.) Find the dimensions x and y that maximize the area given that the perimeter is 100"
The figure is a rectangle with sides x and y with a half-circle connected to one of the ends (diameter of the circle is equal to y).
I got A=xy + 1/2*(1/2y)^2 for the area and
P=2x + y + (1/2)(y)(pi) for the perimeter.
The book gives the answer for c.) as x=100/(4+pi) and y=200/(4+pi)
No matter what I try I can't seem to get their answer. I know I have find the value of x (or y) in terms of the other variable using 100 but I keep getting y=63.661977 and x=31.83098869. I'm stumped.
Hello, Kjcube!
The book's answer are correct.
(a) Find a formula for the area.Code:* * * * * * * * * | * * | * * | y/2 * | * * | * * * * y * | * | * * | y/2 * * | * * | * * * * * * * * * : - - - x - - - :
(b) Find a formula for the perimeter.
(c) Find the dimensions x and y that maximize the area
. . given that the perimeter is 100 inches.
The book gives the answers for (c) as: .
(a) Area = (area of rectangle) + (area of semicircle)
. . .[1]
(b) Perimeter = (perimeter of rectangle) + (perimeter of semicircle)
. .
(c) The perimeter is 100 inches: .
. . Solve for .[2]
Substitute into [1]: .
. . which simplifies to: .
Then: .
Substitute into [2]: .
. . which simplifies to: .