Question

if the central angle of a sector is 2π/3 radians and the radius is 8 meters, what is the sector area?
a. 64π/3 square meters
b. 32π square meters
c. 16π square meters
d. 26π/3 square meters

a bicycle wheel has a radius of 20 cm. if the wheel rotates through an angle of π/6 radians, what is the approximate distance traveled by a point on the outer edge of the wheel?
a. 40 cm
b. 10 cm
c. 20 cm
d. 30 cm

if a central angle measures 2π/5 radians in a circle with radius 5 cm, what is the arc length?
a. 5π cm
b. 2π cm
c. 10π cm
d. 3π cm

Explanation:

Step1: Recall sector - area formula

The formula for the area of a sector of a circle is $A=\frac{1}{2}r^{2}\theta$, where $r$ is the radius and $\theta$ is the central - angle in radians. Given $r = 8$ m and $\theta=\frac{2\pi}{3}$.

Step2: Substitute values into formula

$A=\frac{1}{2}\times8^{2}\times\frac{2\pi}{3}=\frac{1}{2}\times64\times\frac{2\pi}{3}=\frac{64\pi}{3}$ square meters.

Step3: Recall arc - length formula

The formula for the arc - length of a circle is $s = r\theta$, where $r$ is the radius and $\theta$ is the central - angle in radians.

Step4: Solve for the first arc - length problem

For the bicycle wheel with $r = 20$ cm and $\theta=\frac{\pi}{6}$, $s=20\times\frac{\pi}{6}=\frac{10\pi}{3}\approx10$ cm.

Step5: Solve for the second arc - length problem

For a circle with $r = 5$ cm and $\theta=\frac{2\pi}{5}$, $s = 5\times\frac{2\pi}{5}=2\pi$ cm.

Answer:

1. A. $64\pi/3$ square meters
2. B. $10$ cm
3. B. $2\pi$ cm