Question 1203907
you want to find the area in terms of x.


perimeter of the circle is equal to 2 * p * x
x is the radius of the the circle
perimeter of the semi-circle is half that = p * x 


perimeter of the rectangle is equal to 2x + 2y 
2x is one length of the rectangle.
y is one width of the rectangle.
only one length of the rectangle is used because the other length is the base of the semi-circle which is internal to the structure and therefore not counted as part of the perimeter.


the perimeter of the window is the perimeter of the semi-circle plus the perimeter of the rectangle.
since the perimeter of the window is equal to 20, you get:
20 = pi * x + 2x + 2y
solve for 2y to get:
2y = 20 - pi * x - 2x
solve for y to get:
y = 10 - pi/2 * x - x *****


the area of the circle is equal to pi * x^2
the area of the semi-circle is equal to pi * x^2 / 2


the area of the rectangle is equal to 2x * y


the area of the window is equal to the area of the semi-circle plus the area of the rectangle which is equal to pi * x^2/2 + 2x * y *****


you have y = 10 - pi/2 * x - x
replace y with that in the formula for the area of the window to get:
area of the window = pi * x^2/2 + 2x * (10 - pi/2 * x - x)


simplify this to get:
area of the window = pi * x^2/2 + 20x - pi * x^2 - 2x^2
that should be your solution.
it is the area of the window in terms of x.


to see if this makes sense, find a value for x that makes the value of y positive.
i chose 2.
it's chosen randomly and tested to make sure the value of y is positive when using it.


when x = 2, the value of y = 10 - pi/2 * x - x which becomes equal to 10 - pi/2 * 2 - 2 which becomes equal to 4.858407346.


when x = 2 and y = 4.858407346, you get:
area of the window = pi * x^2/2 + 2x * y which becomes equal to pi * 2^2/2 + 2 * 2 * 4.858407346 which is equal to 25.71681469


to get the formula for the area of the window in in terms of x, replace y with (10 - pi/2 * x - x) to get:
the area of the window = pi * x^2/2 + 2x * (10 - pi/2 * x - x) which becomes equal to pi * 2^2/2 + 2*2 * (10 - pi/2 * 2 - 2) which is equal to 25.71681469.


this confirms thqt the formula for the area of the window in terms of x is correct.


the formula is area of the window = pi * x^2/2 + 2x * (10 - pi/2 * x - x) .
this can be simplified to:
area of the window = 20x - 2x^2 - pi/2 * x^2
if you combine like terms, the formula becomes:
area of the window = 20x - (2 + pi/2) * x^2


i checked the numbers several times and i think i finaly got it down right.
the answer should be:
area of the window equals:
20x - (2 + pi/2) * x^2 or:
20x - 2x^2 - pi/2 * x^2