Question

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

how many radians are there in a quarter of a circle?
a. π/3 radians
b. π/6 radians
c. π/2 radians
d. π/8 radians

Explanation:

Step1: Recall the formula for arc length

The formula for arc length \(s\) is \(s = r\theta\), where \(r\) is the radius and \(\theta\) is the central angle in radians.
Given \(r = 5\mathrm{cm}\) and \(\theta=\frac{2\pi}{5}\) radians.

Step2: Substitute values into the formula

\(s=5\times\frac{2\pi}{5}\)

Step3: Simplify the expression

\(s = 2\pi\mathrm{cm}\)

Step4: Recall the total radians in a circle

A full - circle has \(2\pi\) radians.
A quarter - circle is \(\frac{1}{4}\) of a full - circle.

Step5: Calculate radians in a quarter - circle

\(\theta=\frac{1}{4}\times2\pi=\frac{\pi}{2}\) radians

Answer:

For the arc - length problem: a. \(2\pi\mathrm{cm}\)
For the radians - in - a - quarter - circle problem: c. \(\frac{\pi}{2}\) radians