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Answer :
Sure! Let's break down the problem step-by-step to find the measure of the central angle for the arc Rob and his brother traveled on the Ferris wheel.
1. Determine the Diameter and Radius of the Ferris wheel:
- The diameter of the Ferris wheel is given as 40 feet.
- The radius [tex]\( r \)[/tex] is half of the diameter:
[tex]\[
r = \frac{40}{2} = 20 \text{ feet}
\][/tex]
2. Calculate the Circumference of the Ferris wheel:
- The circumference [tex]\( C \)[/tex] of a circle is calculated using the formula [tex]\( C = 2\pi r \)[/tex]:
[tex]\[
C = 2 \pi \times 20 = 40 \pi \text{ feet}
\][/tex]
3. Given Distance Traveled:
- Rob and his brother traveled a distance of [tex]\( \frac{86}{3} \pi \)[/tex] feet.
4. Determine the Proportion of the Circle Traveled:
- This proportion is the fraction of the circumscribed distance (circumference) that they've traveled:
[tex]\[
\text{Proportion} = \frac{\text{Distance Traveled}}{\text{Circumference}} = \frac{\frac{86}{3} \pi}{40 \pi}
\][/tex]
5. Simplifying the Proportion:
- Notice that [tex]\( \pi \)[/tex] cancels out in the numerator and the denominator:
[tex]\[
\text{Proportion} = \frac{\frac{86}{3}}{40} = \frac{86}{3 \times 40} = \frac{86}{120} = \frac{43}{60}
\][/tex]
6. Convert this Proportion to the Central Angle in Degrees:
- Since the entire circumference corresponds to 360 degrees, the central angle for the proportion of the circle traveled is:
[tex]\[
\text{Central Angle} = \left(\frac{43}{60}\right) \times 360
\][/tex]
7. Calculate the Central Angle:
- Simplify the multiplication:
[tex]\[
\text{Central Angle} = \left(\frac{43}{60}\right) \times 360 = \left(\frac{43 \times 360}{60}\right) = 43 \times 6 = 258
\][/tex]
Therefore, the measure of the central angle for the arc they traveled is:
[tex]\[
\boxed{258}
\][/tex]
1. Determine the Diameter and Radius of the Ferris wheel:
- The diameter of the Ferris wheel is given as 40 feet.
- The radius [tex]\( r \)[/tex] is half of the diameter:
[tex]\[
r = \frac{40}{2} = 20 \text{ feet}
\][/tex]
2. Calculate the Circumference of the Ferris wheel:
- The circumference [tex]\( C \)[/tex] of a circle is calculated using the formula [tex]\( C = 2\pi r \)[/tex]:
[tex]\[
C = 2 \pi \times 20 = 40 \pi \text{ feet}
\][/tex]
3. Given Distance Traveled:
- Rob and his brother traveled a distance of [tex]\( \frac{86}{3} \pi \)[/tex] feet.
4. Determine the Proportion of the Circle Traveled:
- This proportion is the fraction of the circumscribed distance (circumference) that they've traveled:
[tex]\[
\text{Proportion} = \frac{\text{Distance Traveled}}{\text{Circumference}} = \frac{\frac{86}{3} \pi}{40 \pi}
\][/tex]
5. Simplifying the Proportion:
- Notice that [tex]\( \pi \)[/tex] cancels out in the numerator and the denominator:
[tex]\[
\text{Proportion} = \frac{\frac{86}{3}}{40} = \frac{86}{3 \times 40} = \frac{86}{120} = \frac{43}{60}
\][/tex]
6. Convert this Proportion to the Central Angle in Degrees:
- Since the entire circumference corresponds to 360 degrees, the central angle for the proportion of the circle traveled is:
[tex]\[
\text{Central Angle} = \left(\frac{43}{60}\right) \times 360
\][/tex]
7. Calculate the Central Angle:
- Simplify the multiplication:
[tex]\[
\text{Central Angle} = \left(\frac{43}{60}\right) \times 360 = \left(\frac{43 \times 360}{60}\right) = 43 \times 6 = 258
\][/tex]
Therefore, the measure of the central angle for the arc they traveled is:
[tex]\[
\boxed{258}
\][/tex]
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