Question

A circle has radius 6 cm. Suppose an arc on the circle has length 31 cm. Answer the

following:
a. The fraction of the whole circle represented by this arc is
b. The measure of the central angle defined by the arc is
degrees.
C. The area of the sector defined by the arc is
cm2. (Express answer
in terms of r or pi).

Answer 1

`Fraction = (31)/(38)`

`\theta = 296.18^\circ`

`Area = 29.62 \pi`

Explanation:

Given

`r = 6`--- radius

`L = 31` --- arc length

Solving (a): Fraction represented by the arc

First, calculate the circumference (C)

`C = 2\pi r`

So, we have:

`C = 2 * 3.14 * 6`

`C = 38` --- approximated

So, the fraction represented by the arc is:

`Fraction = (L)/(C)`

`Fraction = (31)/(38)`

Solving (b): Measure of the center angle

Using arc length formula, we have:

`L = (\theta)/(360) * 2\pi r`

This gives

`31 = (\theta)/(360) * 2 * 3.14 * 6`

`31 = (\theta)/(360) * 37.68`

Make
`\theta` the subject

`\theta = (360 * 31)/(37.68)`

`\theta = 296.18^\circ`

Solving (c): The area of the sector

This is calculated as:

`Area = (\theta)/(360) * \pi r^2`

So, we have:

`Area = (296.18)/(360) * \pi * 6^2`

`Area = (296.18)/(360) * \pi * 36`

Divide 360 and 36

`Area = (296.18)/(10) * \pi`

`Area = 29.618 * \pi\\`

`Area = 29.62 \pi` --- approximated