Respuesta :
We are asked to determine an equation for the angle of a sector as a function of its radius. To do that let's remember that we have the following relationship:
[tex]s=r\theta[/tex]Where "s" is the measurement of the arc of the sector and "r" its radius. Solving for the angle we get:
[tex]\theta=\frac{s}{r}[/tex]Since we are given the perimeter, we can use the formula for the perimeter of a circular sector:
[tex]P=2r+s[/tex]Solving for "s":
[tex]P-2r=s[/tex]Replacing the value of the perimeter:
[tex]40-2r=s[/tex]Replacing the value of "s" in the formula for the angle:
[tex]\theta=\frac{40-2r}{r}[/tex]This is the formula for the angle of the sector as a function of the radius.
To find the formula for the area, let's remember that the area of a circular sector is given by the following equation:
[tex]A=\frac{1}{2}r^2\theta[/tex]Replacing the value of the angle we get:
[tex]A=\frac{1}{2}r^2(\frac{40-2r}{r})[/tex]Simplifying:
[tex]A=r(20-r)[/tex]