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Recall the formula [tex]\(\frac{\text{Arc length}}{\text{Circumference}} = \frac{\theta}{360^\circ}\)[/tex].

Choose the correct answer:
A. [tex]\(98 \pi\)[/tex] inches
B. [tex]\(124 \pi\)[/tex] inches
C. [tex]\(144 \pi\)[/tex] inches
D. [tex]\(288 \pi\)[/tex] inches


Sagot :

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