Optimization of a rectangular window surmounted on a semicircle

[Differentiate]

Homework Statement

A decorative window has the form of a rectangle surmounted by a semicircle whose diameter is equal to the top of the rectangle. If the TOTAL perimeter of the window 16+pi, then what is the maximum area?

A. 25.653
B. 32.148
C. 15.923
D. 38.047
E. 30.018

Correct answer is A: 25.653, but explain step by step please.

2. The attempt at a solution

I completely started off on the wrong foot here.
What I did was made the radius = x/2 where x is the total width/diameter of the rectangle/circle. Then I made the equations:

P=2∏(x/2)+2x+2y=16+pi
A=∏(x/2)^2+xy=z

I seem to not be getting the answer after I plug everything in, so I know I am starting off wrong.
Please explain by a step-step process.
Thanks in advance.

 

[SteamKing]

Eliminate some of your unknowns by assuming that the rectangle has a certain ratio of width to height, so that x = r*y. Then the perimeter can be expressed as a function of x, which can be substituted into the area formula.

 

[Ray Vickson]

Homework Statement

A decorative window has the form of a rectangle surmounted by a semicircle whose diameter is equal to the top of the rectangle. If the TOTAL perimeter of the window 16+pi, then what is the maximum area?

A. 25.653
B. 32.148
C. 15.923
D. 38.047
E. 30.018

Correct answer is A: 25.653, but explain step by step please.

2. The attempt at a solution

I completely started off on the wrong foot here.
What I did was made the radius = x/2 where x is the total width/diameter of the rectangle/circle. Then I made the equations:

P=2∏(x/2)+2x+2y=16+pi
A=∏(x/2)^2+xy=z

I seem to not be getting the answer after I plug everything in, so I know I am starting off wrong.
Please explain by a step-step process.
Thanks in advance.

Your formulas for P and A are wrong. Draw a carefully-labelled diagram, showing x, y, x/2, etc., and then see where your error lies.