Please Help D:. Min/Max Problems

Question

4. Joan has 50ft of interlocking stone available of fencing off a flower garden in the form of a circular sector. Find the radius of the circle that will yield a flower garden with the largest are if Joan uses all the stone.

Answer 1

The perimeter of the garden is

p = 2r + rθ, so
50 = r(2+θ), and θ = 50/r - 2

the area is
a = 1/2 r^2 θ
= 1/2 r^2 (50/r - 2)
= 25r - r^2
= r(25-r)

This is just a parabola with vertex at r=25/2, so that's the radius giving maximum area