Answer :
Let's solve the problem step-by-step using the given formula. The formula to find the circumference [tex]\( C \)[/tex] from the arc length [tex]\( s \)[/tex] is given by:
[tex]\[ \frac{\text{Arc length}}{\text{Circumference}} = \frac{\theta}{360^\circ} \][/tex]
where [tex]\(\theta\)[/tex] is the central angle in degrees.
Given that [tex]\( \theta = 360^\circ \)[/tex], the formula simplifies to:
[tex]\[ \frac{\text{Arc length}}{\text{Circumference}} = \frac{360^\circ}{360^\circ} = 1 \][/tex]
Hence,
[tex]\[ \text{Circumference} = \frac{\text{Arc length} \times 360^\circ}{\pi^\circ} \][/tex]
Now, we will use this formula to calculate the circumference for each arc length.
### For the arc length [tex]\( 98\pi \)[/tex] inches:
[tex]\[ \text{Circumference} = \frac{98\pi \times 360}{\pi} = 98 \times 360 = 35280 \, \text{inches} \][/tex]
### For the arc length [tex]\( 124\pi \)[/tex] inches:
[tex]\[ \text{Circumference} = \frac{124\pi \times 360}{\pi} = 124 \times 360 = 44640 \, \text{inches} \][/tex]
### For the arc length [tex]\( 144\pi \)[/tex] inches:
[tex]\[ \text{Circumference} = \frac{144\pi \times 360}{\pi} = 144 \times 360 = 51840 \, \text{inches} \][/tex]
### For the arc length [tex]\( 288\pi \)[/tex] inches:
[tex]\[ \text{Circumference} = \frac{288\pi \times 360}{\pi} = 288 \times 360 = 103680 \, \text{inches} \][/tex]
Therefore, the circumferences corresponding to the given arc lengths are:
1. Arc length [tex]\(98\pi\)[/tex] inches: Circumference = [tex]\(35280\)[/tex] inches
2. Arc length [tex]\(124\pi\)[/tex] inches: Circumference = [tex]\(44640\)[/tex] inches
3. Arc length [tex]\(144\pi\)[/tex] inches: Circumference = [tex]\(51840\)[/tex] inches
4. Arc length [tex]\(288\pi\)[/tex] inches: Circumference = [tex]\(103680\)[/tex] inches
[tex]\[ \frac{\text{Arc length}}{\text{Circumference}} = \frac{\theta}{360^\circ} \][/tex]
where [tex]\(\theta\)[/tex] is the central angle in degrees.
Given that [tex]\( \theta = 360^\circ \)[/tex], the formula simplifies to:
[tex]\[ \frac{\text{Arc length}}{\text{Circumference}} = \frac{360^\circ}{360^\circ} = 1 \][/tex]
Hence,
[tex]\[ \text{Circumference} = \frac{\text{Arc length} \times 360^\circ}{\pi^\circ} \][/tex]
Now, we will use this formula to calculate the circumference for each arc length.
### For the arc length [tex]\( 98\pi \)[/tex] inches:
[tex]\[ \text{Circumference} = \frac{98\pi \times 360}{\pi} = 98 \times 360 = 35280 \, \text{inches} \][/tex]
### For the arc length [tex]\( 124\pi \)[/tex] inches:
[tex]\[ \text{Circumference} = \frac{124\pi \times 360}{\pi} = 124 \times 360 = 44640 \, \text{inches} \][/tex]
### For the arc length [tex]\( 144\pi \)[/tex] inches:
[tex]\[ \text{Circumference} = \frac{144\pi \times 360}{\pi} = 144 \times 360 = 51840 \, \text{inches} \][/tex]
### For the arc length [tex]\( 288\pi \)[/tex] inches:
[tex]\[ \text{Circumference} = \frac{288\pi \times 360}{\pi} = 288 \times 360 = 103680 \, \text{inches} \][/tex]
Therefore, the circumferences corresponding to the given arc lengths are:
1. Arc length [tex]\(98\pi\)[/tex] inches: Circumference = [tex]\(35280\)[/tex] inches
2. Arc length [tex]\(124\pi\)[/tex] inches: Circumference = [tex]\(44640\)[/tex] inches
3. Arc length [tex]\(144\pi\)[/tex] inches: Circumference = [tex]\(51840\)[/tex] inches
4. Arc length [tex]\(288\pi\)[/tex] inches: Circumference = [tex]\(103680\)[/tex] inches
