The total perimeter of a sector is 2. Find an expression for its area in terms of:
a) Its Radius
b) Its Angle
I tried solving for r using the perimeter equation,
i.e. 2 = 2(r) + rØ
r = 2/(2+Ø)
And then I subbed that into the area equation,
i.e. A = (1/2)(2/2+ø)^2(ø)
And then I get
A = (2/(2+Ø)^2(Ø)
But the answer on the memo is:
a) r(1-r)
b) 2Ø/(2+Ø)^2
$2=2r+r\theta \implies \theta=\dfrac{2(1-r)}{r}$
$A=\dfrac{r^2}{2} \cdot \theta = \dfrac{r^2}{2} \cdot \dfrac{2(1-r)}{r} = r(1-r)$
$2=r(2+\theta) \implies r=\dfrac{2}{2+\theta}$
$A=\dfrac{r^2}{2} \cdot \theta = \dfrac{1}{2} \cdot \dfrac{2^2}{(2+\theta)^2} \cdot \theta = \dfrac{2\theta}{(2+\theta)^2}$