Question

the diameter of a circle is 10 kilometers. what is the angle measure of an arc bounding a sector with area 5π square kilometers?
k=5π sq. km

d=10 km

give the exact answer in simplest form.

°
π
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work it out

Explanation:

Step1: Find the radius of the circle

The radius \(r\) of a circle is half of the diameter \(d\). Given \(d = 10\) km, so \(r=\frac{d}{2}=\frac{10}{2}=5\) km.

Step2: Use the sector - area formula

The formula for the area of a sector of a circle is \(K=\frac{\theta}{360^{\circ}}\times\pi r^{2}\), where \(K\) is the area of the sector, \(\theta\) is the central angle (in degrees), and \(r\) is the radius of the circle.
We know \(K = 5\pi\) and \(r = 5\). Substitute these values into the formula:

$$5\pi=\frac{\theta}{360^{\circ}}\times\pi\times(5)^{2}$$

First, cancel out \(\pi\) on both sides of the equation:

$$5=\frac{\theta}{360^{\circ}}\times25$$

Step3: Solve for \(\theta\)

Multiply both sides of the equation \(5=\frac{\theta}{360^{\circ}}\times25\) by \(360^{\circ}\):

$$5\times360^{\circ}=25\theta$$

Then, divide both sides by 25:

$$\theta=\frac{5\times360^{\circ}}{25}$$

$$=\frac{1800^{\circ}}{25}=72^{\circ}$$

Answer:

\(72\)