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Answer :
To find the measure of the central angle for the arc that Rob and his brother traveled on the Ferris wheel, we will use the information given and a few formulas.
### Step-by-step Solution:
1. Understand the Relationship Between Circumference and Central Angle:
- The distance traveled along a circular path (arc length) relates to the central angle of the circle. The formula to find the central angle in degrees is:
[tex]\[
\text{Central Angle (degrees)} = \left( \frac{\text{Arc Length}}{\text{Circumference}} \right) \times 360
\][/tex]
2. Find the Circumference of the Ferris Wheel:
- The Ferris wheel forms a circle with a diameter of 40 feet.
- The formula for the circumference [tex]\( C \)[/tex] of a circle is:
[tex]\[
C = \pi \times \text{Diameter} = \pi \times 40 \text{ feet}
\][/tex]
3. Compute the Central Angle:
- The arc length (distance traveled) is given as [tex]\(\frac{86}{3} \pi\)[/tex] feet.
- Plug the arc length and circumference into the formula for the central angle:
[tex]\[
\text{Central Angle (degrees)} = \left( \frac{\frac{86}{3} \pi}{40 \pi} \right) \times 360
\][/tex]
- Simplifying the expression:
- The [tex]\(\pi\)[/tex] in the numerator and denominator cancels out.
- The fraction becomes [tex]\(\frac{86}{120}\)[/tex], which simplifies further.
4. Calculate:
- Simplify the fraction and multiply by 360 to find the central angle in degrees:
[tex]\[
\text{Central Angle (degrees)} = \frac{86}{120} \times 360 = 258
\][/tex]
Thus, the measure of the associated central angle for the arc traveled by Rob and his brother is 258 degrees.
### Step-by-step Solution:
1. Understand the Relationship Between Circumference and Central Angle:
- The distance traveled along a circular path (arc length) relates to the central angle of the circle. The formula to find the central angle in degrees is:
[tex]\[
\text{Central Angle (degrees)} = \left( \frac{\text{Arc Length}}{\text{Circumference}} \right) \times 360
\][/tex]
2. Find the Circumference of the Ferris Wheel:
- The Ferris wheel forms a circle with a diameter of 40 feet.
- The formula for the circumference [tex]\( C \)[/tex] of a circle is:
[tex]\[
C = \pi \times \text{Diameter} = \pi \times 40 \text{ feet}
\][/tex]
3. Compute the Central Angle:
- The arc length (distance traveled) is given as [tex]\(\frac{86}{3} \pi\)[/tex] feet.
- Plug the arc length and circumference into the formula for the central angle:
[tex]\[
\text{Central Angle (degrees)} = \left( \frac{\frac{86}{3} \pi}{40 \pi} \right) \times 360
\][/tex]
- Simplifying the expression:
- The [tex]\(\pi\)[/tex] in the numerator and denominator cancels out.
- The fraction becomes [tex]\(\frac{86}{120}\)[/tex], which simplifies further.
4. Calculate:
- Simplify the fraction and multiply by 360 to find the central angle in degrees:
[tex]\[
\text{Central Angle (degrees)} = \frac{86}{120} \times 360 = 258
\][/tex]
Thus, the measure of the associated central angle for the arc traveled by Rob and his brother is 258 degrees.
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