contestada

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

Respuesta :

sqdancefan
The area of the circular sector is
[tex]A=\frac{1}{2}r^{2}\theta[/tex]

From that, the equation for the central angle is
[tex]\theta=\dfrac{2A}{r^{2}}[/tex]

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

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

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

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.

Otras preguntas