To determine the measure of the central angle corresponding to the sector [tex]\( \widehat{A B} \)[/tex] in radians, let's go through the following steps:
1. Understand the problem:
- We are given that the ratio of the area of sector [tex]\( AOB \)[/tex] to the area of the entire circle is [tex]\( \frac{3}{5} \)[/tex].
- Our goal is to find the measure of the central angle, [tex]\(\theta\)[/tex], in radians that corresponds to the sector [tex]\( AOB \)[/tex].
2. Relate sector area to central angle:
- The area of a sector of a circle is given by [tex]\(\frac{\theta}{2\pi} \times \pi r^2\)[/tex] where [tex]\(r\)[/tex] is the radius of the circle and [tex]\(\theta\)[/tex] is the central angle in radians.
- The area of the whole circle is [tex]\(\pi r^2\)[/tex].
3. Set up the ratio:
- Given that the ratio of the area of the sector to the area of the circle is [tex]\(\frac{3}{5}\)[/tex], we can write the equation:
[tex]\[
\frac{\frac{\theta}{2\pi} \times \pi r^2}{\pi r^2} = \frac{3}{5}
\][/tex]
4. Simplify the equation:
- The [tex]\(\pi r^2\)[/tex] terms cancel out, leaving:
[tex]\[
\frac{\theta}{2\pi} = \frac{3}{5}
\][/tex]
5. Solve for [tex]\(\theta\)[/tex]:
- Multiply both sides by [tex]\(2\pi\)[/tex] to isolate [tex]\(\theta\)[/tex]:
[tex]\[
\theta = \left(\frac{3}{5}\right) \times 2\pi
\][/tex]
6. Calculate [tex]\(\theta\)[/tex]:
- Substituting the value of [tex]\(\pi\)[/tex]:
[tex]\[
\theta = \left(\frac{3}{5}\right) \times 2 \times \pi \approx \left(\frac{3}{5}\right) \times 2 \times 3.14159
\][/tex]
Performing the multiplication:
[tex]\[
\theta \approx \left(\frac{3}{5}\right) \times 6.28318 \approx 3.7699111843077517
\][/tex]
7. Round to two decimal places:
- The approximate measure of the central angle [tex]\(\theta\)[/tex], rounded to two decimal places, is:
[tex]\[
\theta \approx 3.77
\][/tex]
8. Select the correct answer:
- From the given options, the closest value rounded to two decimal places is:
[tex]\[
\boxed{3.77}
\][/tex]
