Rectangular auditorium optimization problem

Question

We wish to construct a rectangular auditorium with a stage shaped as a semicircle of radius $r$, as shown in the diagram below (white is the stage and green is the seating area). For safety reasons, light strips must be placed on the perimeter of the seating area. If we have $45\pi + 60$ meters of light strips, what should $r$ be so that the seating area is maximized?

My issue is that I tried to solve it the "geometric way", figuring that the perimeter (in r) for the seating area is going to be $(3+\pi)r$. I set it equal to $45\pi+60$ to try and find the closest number to equal the value of $r$. Is this the way I should approach the problem, or is there a "calculus" style method I should use to come to a solution?

Edit:

Answer 1

Hint:   the width of the sitting area is $\,2r\,$, and let its depth be $\,x\,$. Then the perimeter is $\,\pi r + 2x + 2r=45 \pi + 60\,$, and the problem asks to maximize its area $\,2rx - \pi r^2 /2\,$.

The perimeter relation gives $\,x = \big(45 \pi + 60 - (\pi+2)r\big)/2\,$, and substituting in the area relation yields a quadratic in $\,r\,$ to be maximized over $\mathbb{R}^+$, which at this point should be straigtforward.

Answer 2

Hint: the rectangle has length $2r$ and width $x$. Then the seating perimeter constraint: $$2x+2r+\pi r=45\pi +60.$$ The objective function is: $$S(x,r)=2xr-\frac{\pi r^2}{2}.$$ Can you handle the rest?