- #1

togame

- 18

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## Homework Statement

I have a sector of a circle with area 12 square meters. If radius r and angle [itex]\theta[/itex] are chosen so that the that perimeter of the sector is the smallest possible, then what is the radius?

## Homework Equations

I have area of sector as [itex]A=\frac{\theta r^2}{2}[/itex] which is 12

and the length of the arc as [itex]L=\theta r[/itex]

## The Attempt at a Solution

my attempt is as follows:

since i am attempting to minimize the circumference, i will need to minimize that function.

[tex]L=\theta r[/tex]

and since i only want one variable, i use the area function and solve in terms of r and replace it in the circumference function.

[tex]12=\frac{\theta r^2}{2}[/tex]

[tex]\theta=\frac{24}{r^2}[/tex]

[tex]C=r\frac{24}{r^2}[/tex]

[tex]C=\frac{24}{r}[/tex]

now i need to take the derivative of this function and solve set equal to 0 to get r

[tex]C\prime=-\frac{24}{r^2}[/tex]

now, this is where i am stuck. since r is squared, this derivative can never be 0, so i believe i am missing a step somewhere or am just confused about how to set it up. any help would be greatly appreciated.