Question

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. 20 cm
b. 30 cm
c. 40 cm
d. 10 cm
what is the central angle of a sector in radians if its area is half the area of the entire circle?
a. 2π radians
b. π radians
c. 2π/3 radians
d. π/2 radians
what is the formula for calculating arc length in a circle?
a. arclength = diameter × centralangle (in degrees)
b. arclength = circumference × centralangle (in degrees)
c. arclength = 2 × radius × centralangle (in radians)
d. arclength = radius × centralangle (in radians)
if the central angle of a sector is 3 radians and the radius is 6 inches, what is the sector area?
a. 27π square inches
b. 18π square inches
c. 18 square inches
d. 54 square inches

Explanation:

Step1: Recall the arc - length formula

The formula for the arc - length \(s\) of a circle is \(s = r\theta\), where \(r\) is the radius and \(\theta\) is the central angle in radians.
Given \(r = 20\mathrm{cm}\) and \(\theta=\frac{\pi}{6}\) radians.

Step2: Substitute the values into the formula

\(s=20\times\frac{\pi}{6}=\frac{10\pi}{3}\approx10.47\mathrm{cm}\approx10\mathrm{cm}\)

Step3: Recall the sector - area formula

The formula for the area \(A\) of a sector is \(A=\frac{1}{2}r^{2}\theta\), where \(r\) is the radius and \(\theta\) is the central angle in radians.
Given \(r = 6\) inches and \(\theta = 3\) radians.

Step4: Substitute the values into the sector - area formula

\(A=\frac{1}{2}\times6^{2}\times3=\frac{1}{2}\times36\times 3=54\) square inches

Step5: Recall the relationship between sector area and central angle

The area of a circle is \(A_{c}=\pi r^{2}\), and the area of a sector is \(A_{s}=\frac{1}{2}r^{2}\theta\).
If \(A_{s}=\frac{1}{2}A_{c}\), then \(\frac{1}{2}r^{2}\theta=\frac{1}{2}\pi r^{2}\). Canceling out \(\frac{1}{2}r^{2}\) (since \(r
eq0\)), we get \(\theta=\pi\) radians

Step6: Recall the arc - length formula (for the formula question)

The formula for arc - length \(s\) with radius \(r\) and central angle \(\theta\) (in radians) is \(s = r\theta\)

Answer:

1. d. \(10\mathrm{cm}\)
2. b. \(\pi\) radians
3. d. \(ArcLength = Radius\times CentralAngle\) (in radians)
4. d. \(54\) square inches