Tthe approximate measure, in radians, of the central angle corresponding to arc AB is 3.77 rad
Calculating the measure of central angle
From the question, we are to calculate the measure of the central angle corresponding to arc AB
From the given information,
The ratio of the area of sector AOB to the area of the circle is 3/5
The area of a sector is given by the formula,
[tex]Area\ of\ a\ sector = \frac{\theta}{360 ^\circ} \times \pi r^{2}[/tex]
Where θ is the central angle
and r is the radius of the circle
The area of a circle is given by the formula,
[tex]Area\ of\ a\ circle = \pi r^{2}[/tex]
∴ [tex]\frac{Area\ of\ a\ sector}{Area\ of\ a\ circle } = \frac{\frac{\theta}{360 ^\circ} \times \pi r^{2}}{ \pi r^{2}}[/tex]
[tex]\frac{Area\ of\ a\ sector}{Area\ of\ a\ circle } = \frac{\theta}{360 ^\circ}[/tex]
From the given information,
[tex]\frac{Area\ of\ a\ sector}{Area\ of\ a\ circle } = \frac{3}{5}[/tex]
Then,
[tex]\frac{\theta}{360^\circ}=\frac{3}{5}[/tex]
[tex]\theta = \frac{3\times 360^\circ}{5}[/tex]
θ = 216°
The measure of the central angle in degrees is 216°.
Now, for the measure of the central angle in radians
The measure of the central angle in radians = [tex]216^\circ \times \frac{\pi}{180}[/tex] rad
The measure of the central angle in radians = 3.77 rad
Hence, the approximate measure, in radians, of the central angle corresponding to arc AB is 3.77 rad
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