Question

arc length and sector area
consider a circle with a radius of 10 centimeters.
what is the length of the arc that measures 60° for the given circle?
a (\frac{5pi}{3}) cm
b (\frac{5pi}{3}) cm
c (\frac{10pi}{3}) cm
d (\frac{10pi}{9}) cm
what is the sector area with an arc that measures 60° for the given circle?
a (\frac{10pi}{3}) cm²
b (\frac{25pi}{3}) cm²

Explanation:

Step1: Recall the arc - length formula

The formula for the arc - length \(L\) of a circle is \(L=\frac{\theta}{360}\times2\pi r\), where \(\theta\) is the central angle in degrees and \(r\) is the radius of the circle.
Given \(r = 10\) cm and \(\theta=60^{\circ}\).

Step2: Substitute the values into the formula

Substitute \(r = 10\) and \(\theta = 60\) into \(L=\frac{\theta}{360}\times2\pi r\).

$$
LATEXBLOCK0
$$

Step3: Recall the sector - area formula

The formula for the area \(A\) of a sector of a circle is \(A=\frac{\theta}{360}\times\pi r^{2}\), where \(\theta\) is the central angle in degrees and \(r\) is the radius of the circle.
Substitute \(r = 10\) and \(\theta = 60\) into \(A=\frac{\theta}{360}\times\pi r^{2}\).

$$
LATEXBLOCK1
$$

Answer:

• Arc length: \(\frac{10\pi}{3}\text{ cm}\) (Option C for arc - length if we assume the options are labeled as in the problem's structure for arc - length part)
• Sector area: \(\frac{50\pi}{3}\text{ cm}^{2}\) (Option A for sector - area if we assume the options are labeled as in the problem's structure for sector - area part)