Question

Matilda is planning a walk around the perimeter of wedge park, which is shaped like a circular wedge, as shown below. the walk around the park is 2.9 miles, and the park has an area of 0.25 square miles. if θ is less than 90 degrees, what is the value of the radius, r? (round your answer to two decimal places.) mi

Answer 1

The area of the circular sector is
$A=\frac{1}{2}r^{2}\theta$

From that, the equation for the central angle is
$\theta=\dfrac{2A}{r^{2}}$

The perimeter of the circular sector is the sum of the arc length and twice the radius.
$P=2r+r\theta\\\\P=r(2+\theta)$

From that, the equation for the central angle is
$\dfrac{P}{r}=2+\theta\\\\ \theta=\dfrac{P}{r}-2$

The value of r can be found by equating these two expressions for θ.
$\dfrac{2A}{r^{2}}=\dfrac{P}{r}-2\\\\2r^{2}-Pr+2A=0\\\\2r^{2}-2.9r+0.5=0$

By any of several means, you can find that r=0.2 or 1.25. The latter value corresponds to the requirement that the central angle be less than 90°. The radius is ...
  1.25 miles.