To determine the measure of the central angle corresponding to [tex]\(\widehat{A B}\)[/tex], we start by understanding the relationship between the area of a sector and the area of the circle.
Given:
- The ratio of the area of sector [tex]\(AOB\)[/tex] to the area of the entire circle is [tex]\(\frac{3}{5}\)[/tex].
We know that:
- The area of a sector of a circle is proportional to the central angle of that sector.
- The total angle in a circle is [tex]\(2\pi\)[/tex] radians.
Thus, the ratio of the area of the sector to the area of the circle is equal to the ratio of the sector's central angle [tex]\(\theta\)[/tex] to [tex]\(2\pi\)[/tex]:
[tex]\[
\frac{\text{Area of sector } AOB}{\text{Area of the circle}} = \frac{\theta}{2\pi}
\][/tex]
Given:
[tex]\[
\frac{\text{Area of sector } AOB}{\text{Area of the circle}} = \frac{3}{5}
\][/tex]
Equating the ratios, we get:
[tex]\[
\frac{3}{5} = \frac{\theta}{2\pi}
\][/tex]
Solving for [tex]\(\theta\)[/tex], we multiply both sides by [tex]\(2\pi\)[/tex]:
[tex]\[
\theta = \left(\frac{3}{5}\right) \cdot 2\pi
\][/tex]
Now we calculate [tex]\(\theta\)[/tex] explicitly:
[tex]\[
\theta = \frac{3}{5} \cdot 2\pi = \frac{6\pi}{5}
\][/tex]
To approximate the value of [tex]\(\theta\)[/tex], we use the given value:
[tex]\[\pi \approx 3.14159\][/tex]
[tex]\[
\theta \approx \frac{6 \cdot 3.14159}{5} \approx 3.7699111843077517
\][/tex]
Rounding the central angle to two decimal places, we get:
[tex]\[
\theta \approx 3.77 \text{ radians}
\][/tex]
Thus, the approximate measure in radians of the central angle corresponding to [tex]\(\widehat{A B}\)[/tex] is:
[tex]\[
\boxed{3.77}
\][/tex]
