Problem: A sector of a circle, of a radius r cm, has a perimeter 200cm. Given that r can vary, find the stationary value of the area of the sector.
Solution: A=2500
Attempt:
Perimeter = rΘ + 2r
200 = rΘ + 2r
200 - 2r = rΘ
100 - r= 1/2 rΘ
100r - r^2 = 1/2 r^2Θ
Area = 100r - r^2
So I this is what I have so far and not sure how to continue.
This is just a parabola with $a<0$. So, the stationary point is reached in the vertex of the parabola, which have coords: $$V\left(-\frac{b}{2a},-\frac{\Delta}{4a}\right)$$ Substituing for $a,b$ and $c$, we have: $$V\left(\frac{1}{2},\frac{100^2}{4}\right)\implies V\left(\frac{1}{2},2500\right)$$ This shows that the area of the stationary point is $2500$.