Question

written exercises
in circle o with radius of length r, central angle poq intercepts (widehat{qp}). find, to the
nearest tenth, the radian measure of (angle poq) for each given value of r and length
of (widehat{qp}).

1. (r = 8) cm, length of (widehat{qp}=12) cm
2. (r = 1) m, length of (widehat{qp}=2) m

4 (r = 10) cm, length of (widehat{qp}=40) cm

Explanation:

Step1: Recall the formula for arc length

The formula for the length of an arc \(s\) is \(s = r\theta\), where \(r\) is the radius of the circle and \(\theta\) is the radian measure of the central angle. We can solve for \(\theta\) by the formula \(\theta=\frac{s}{r}\).

Step2: Substitute the given values for problem 1

Given \(r = 8\mathrm{cm}\) and \(s=12\mathrm{cm}\). Substitute into \(\theta=\frac{s}{r}\), we get \(\theta=\frac{12}{8}=1.5\) radians.

Step3: Substitute the given values for problem 2

Given \(r = 1\mathrm{m}\) and \(s = 2\mathrm{m}\). Substitute into \(\theta=\frac{s}{r}\), we get \(\theta=\frac{2}{1}=2.0\) radians.

Step4: Substitute the given values for problem 4 (assuming it was a typo and you meant the third problem)

Given \(r=10\mathrm{cm}\) and \(s = 40\mathrm{cm}\). Substitute into \(\theta=\frac{s}{r}\), we get \(\theta=\frac{40}{10} = 4.0\) radians.

Answer:

1. \(1.5\) radians
2. \(2.0\) radians
3. \(4.0\) radians