A window is in the form of a rectangle surmounted by a semicircle.
The rectangle is made of clear glass, whereas the semicircle is of tinted glass that transmits only half as much light per unit area as the clear glass does. The total perimeter is fixed.
Find the proportions of the window that will admit the most light. Neglect that thickness of the frame.
I realize that the total light will be equal to how much light the surface transmits times the amount of area.
If you call the base of the rectangle $x$, height $y$, and the light transmitted by clear glass $L$ you can say that total light is equal to $x\cdot y\cdot L + \frac{1}{2}\cdot\pi\cdot x^2\cdot\frac{1}{2}L$
This seems like too many variable for an optimization problem. How should I proceed?