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Answer :
To find the measure of the central angle for the arc Rob and his brother traveled on the Ferris wheel, we can follow these steps:
1. Understand the Problem: The Ferris wheel has a diameter of 40 feet, and they traveled an arc distance of [tex]\(\frac{86}{3} \pi\)[/tex] feet.
2. Calculate the Circumference of the Ferris Wheel:
- The formula for the circumference [tex]\(C\)[/tex] of a circle is:
[tex]\[
C = \pi \times \text{diameter}
\][/tex]
- With a diameter of 40 feet, the circumference becomes:
[tex]\[
C = \pi \times 40 = 40\pi \text{ feet}
\][/tex]
3. Determine the Fraction of the Circumference That the Arc Represents:
- You have an arc length of [tex]\(\frac{86}{3} \pi\)[/tex] feet.
- To find what fraction of the total circumference this is, divide the arc length by the circumference:
[tex]\[
\frac{\frac{86}{3} \pi}{40 \pi} = \frac{86}{120} = \frac{43}{60}
\][/tex]
- This fraction represents the portion of the Ferris wheel's total circumference that they traveled.
4. Calculate the Central Angle:
- The central angle in degrees can be calculated by multiplying this fraction by 360 degrees (since a full circle is 360 degrees):
[tex]\[
\text{Central Angle} = \frac{43}{60} \times 360 = 258 \text{ degrees}
\][/tex]
So, the measure of the associated central angle for the arc traveled is [tex]\(258^\circ\)[/tex].
1. Understand the Problem: The Ferris wheel has a diameter of 40 feet, and they traveled an arc distance of [tex]\(\frac{86}{3} \pi\)[/tex] feet.
2. Calculate the Circumference of the Ferris Wheel:
- The formula for the circumference [tex]\(C\)[/tex] of a circle is:
[tex]\[
C = \pi \times \text{diameter}
\][/tex]
- With a diameter of 40 feet, the circumference becomes:
[tex]\[
C = \pi \times 40 = 40\pi \text{ feet}
\][/tex]
3. Determine the Fraction of the Circumference That the Arc Represents:
- You have an arc length of [tex]\(\frac{86}{3} \pi\)[/tex] feet.
- To find what fraction of the total circumference this is, divide the arc length by the circumference:
[tex]\[
\frac{\frac{86}{3} \pi}{40 \pi} = \frac{86}{120} = \frac{43}{60}
\][/tex]
- This fraction represents the portion of the Ferris wheel's total circumference that they traveled.
4. Calculate the Central Angle:
- The central angle in degrees can be calculated by multiplying this fraction by 360 degrees (since a full circle is 360 degrees):
[tex]\[
\text{Central Angle} = \frac{43}{60} \times 360 = 258 \text{ degrees}
\][/tex]
So, the measure of the associated central angle for the arc traveled is [tex]\(258^\circ\)[/tex].
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