Answer:
Step-by-step explanation:
The answer is definitely not 65.
This problem requires the following formulas:
Area of a circle = πr²,
Arc Length = [tex]\frac{\theta}{360}*2\pi r[/tex], and
Area of a sector = [tex]\frac{\theta}{360}*\pi r^2[/tex]
Notice that each of these has a radius in it.
That means that first you have to find the length of the radius in order to be able to solve for anything else. We will use the area of a circle to find the radius. We cannot use either one of the other 2 since the radius is still unknown and so is the central angle theta. The area of the circle is given as 201.06. Therefore,
[tex]201.06=\pi r^2[/tex]
Divide both sides by the value for π to get:
[tex]63.99938572=r^2[/tex] and take the square root of both sides to get that the radius is
r = 7.9999 so we will round to 8.0 (to the nearest tenth).
Now that we know that the radius is 8, we can use that to find the measure of the central angle theta in the area of a sector formula. We know that the area of the shaded sector is 75.40, therefore,
[tex]75.40=\frac{\theta}{360}*\pi (8^2)[/tex] and simplifying:
[tex]75.40=\frac{64\pi \theta}{360}[/tex]
Multiply both sides by 360 to get
27144 = 64πθ, and divide both sides by 64π to get that
θ = 135°. Remember that the measure of the central angle is also the arc measure in degrees. So the arc measure is also 135°.
Now that we know theta we can find the arc length:
[tex]AL=\frac{135}{360}*16\pi[/tex], and simplifying a bit:
[tex]AL=.375(16\pi)[/tex] so the
Arc Length is 18.8 cm
