The measure of the central angle for the group that likes the program would be of 2 radians approx
What is the relation between angle subtended by the arc, the radius and the arc length?
[tex]2\pi^c = 360^\circ = \text{Full circumference}[/tex]
The superscript 'c' shows angle measured is in radians.
If radius of the circle is of r units, then:
[tex]1^c \: \rm covers \: \dfrac{circumference}{2\pi} = \dfrac{2\pi r}{2\pi} = r\\\\or\\\\\theta^c \: covers \:\:\: r \times \theta \: \rm \text{units of arc}[/tex]
Now, for this case, the circle graph(referring to what is often called pie graph) would be such that:
190 students, 135 students and 220 students will fill the full circle.
We will use the same concept as discussed above.
Full circle subtends [tex]2\pi[/tex] radians on the center.
The full arc is of [tex]2\pi r[/tex] units length.
Total students are 190 + 135 + 220 = 545 students.
Each student assumingly taking up uniform space in area of circle, thus also dividing whole circle in 545 equal slices, from center to circumference.
Thus, each arc would be of length [tex]2\pi r/545[/tex] units length.
Then that means 190 students (the count of students liking the program) would have the length of the arc covered as [tex]190 \times \dfrac{2\pi r}{545}[/tex] units.
Let this makes angle [tex]\theta[/tex] radians on center. Then, we have:
Length of the arc = [tex]r \times \theta[/tex] units
[tex]190 \times \dfrac{2\pi r}{545} = r \times \theta\\\\\theta = \dfrac{380\pi}{545} \approx 2.1904 \: \rm (in \: \rm radians) \approx 2^c[/tex]
Thus, the measure of the central angle for the group that likes the program would be of 2 radians approx
Learn more about arc and angle relation here:
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