Let's go through each question step-by-step using the formula [tex]\( s = r \theta \)[/tex], where [tex]\( s \)[/tex] is the length of the circular arc, [tex]\( r \)[/tex] is the radius, and [tex]\( \theta \)[/tex] is the central angle in radians.
(a) Find the length [tex]\( s \)[/tex] of a circular arc when [tex]\( r = 9 \)[/tex] and [tex]\( \theta = \frac{5 \pi}{6} \)[/tex].
Given:
[tex]\[ r = 9 \][/tex]
[tex]\[ \theta = \frac{5 \pi}{6} \][/tex]
Using the formula [tex]\( s = r \theta \)[/tex]:
[tex]\[ s = 9 \times \frac{5 \pi}{6} \][/tex]
After substituting the values:
[tex]\[ s \approx 23.56194490192345 \][/tex]
(b) Find the length [tex]\( s \)[/tex] of a circular arc when [tex]\( r = 5 \)[/tex] and [tex]\( \theta = 140^{\circ} \)[/tex].
First, convert the angle from degrees to radians.
[tex]\[ \theta = 140^{\circ} \times \left( \frac{\pi}{180^{\circ}} \right) \][/tex]
[tex]\[ \theta \approx 140 \times 0.0174533 \][/tex]
[tex]\[ \theta \approx 2.44346 \text{ radians} \][/tex]
Now, using the formula [tex]\( s = r \theta \)[/tex]:
[tex]\[ s = 5 \times 2.44346 \][/tex]
[tex]\[ s \approx 12.217304763960307 \][/tex]
(c) Find the length [tex]\( s \)[/tex] of a circular arc when [tex]\( r = 5 \)[/tex] and [tex]\( \theta = 3 \)[/tex] rad.
Given:
[tex]\[ r = 5 \][/tex]
[tex]\[ \theta = 3 \text{ radians} \][/tex]
Using the formula [tex]\( s = r \theta \)[/tex]:
[tex]\[ s = 5 \times 3 \][/tex]
[tex]\[ s = 15 \][/tex]
(d) Find the length [tex]\( s \)[/tex] of a circular arc when [tex]\( r = 12 \)[/tex] and [tex]\( \theta = 40^{\circ} \)[/tex].
First, convert the angle from degrees to radians.
[tex]\[ \theta = 40^{\circ} \times \left( \frac{\pi}{180^{\circ}} \right) \][/tex]
[tex]\[ \theta \approx 40 \times 0.0174533 \][/tex]
[tex]\[ \theta \approx 0.698132 \text{ radians} \][/tex]
Now, using the formula [tex]\( s = r \theta \)[/tex]:
[tex]\[ s = 12 \times 0.698132 \][/tex]
[tex]\[ s \approx 8.377580409572781 \][/tex]
So, the lengths of the circular arcs are:
1. (a) [tex]\( s \approx 23.56194490192345 \)[/tex]
2. (b) [tex]\( s \approx 12.217304763960307 \)[/tex]
3. (c) [tex]\( s = 15 \)[/tex]
4. (d) [tex]\( s \approx 8.377580409572781 \)[/tex]
