This is a 2 parts problem.
- Find (using one variable integrals) the area of a circular sector of raidius r and angle $\theta \leq\pi$.
So for this i used the expression:
$x^2 +y^2 = r^2$ $\Rightarrow$ $y=\sqrt{r^2-x^2}$
Taking the postivie root because of the angles i'm working with. Then, doing some calculation i ended up with the result $A=\frac{1}{2}\pi r^2$, being consistent with what i needed to find because it has to be half the area of a circle of raidius r.
The second part of the problem is where i don't know how to preceed.
- A flower bed in the form of cirucular sector of radius r and $\theta \leq\pi$ will be made. The area is stipulated, it must be $A$. Find r and $\theta$ so that the perimeter of the flower bed is minimal.
So here, i don't know how to proceed as i don't know with which function i should be working, and it seems like a multivariable optimization problem, which is not the objective of the problem. Any hint? Is my original function wrong ?
$\frac12\pi r^2$ isn't a function of $\theta$, so it comes out that the area is constant no matter what $\theta$ is -- which can't be right. (You're close though!)
As for converting a two-variable optimization problem into a one-variable optimization problem: you are optimizing perimeter, so you'll be taking the derivative of the perimeter function. To convert the perimeter function of $r$ and $\theta$ into a function of one variable only, use the relationship between $r$ and $\theta$ determined by the area function (the function which is being held constant).