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Answer :
To solve this problem, we need to determine the measure of the central angle corresponding to the distance Rob and his brother traveled on the Ferris wheel.
1. Diameter of the Ferris Wheel:
The diameter of the Ferris wheel is given as 40 feet.
2. Circumference Calculation:
The circumference of the Ferris wheel can be calculated using the formula [tex]\( C = \pi \times \text{diameter} \)[/tex].
[tex]\[ C = \pi \times 40 \][/tex]
3. Distance Traveled:
They traveled a distance of [tex]\( \frac{86}{3} \pi \)[/tex] feet.
4. Fraction of the Circumference Traveled:
To find the fraction of the circumference they traveled, divide the distance traveled by the circumference:
[tex]\[ \text{Fraction} = \frac{\text{Distance Traveled}}{\text{Circumference}} = \frac{\left(\frac{86}{3} \pi\right)}{\pi \times 40} \][/tex]
5. Simplify the Fraction:
When simplifying the fraction, the [tex]\(\pi\)[/tex] terms cancel out:
[tex]\[ \text{Fraction} = \frac{86}{120} = \frac{43}{60} \][/tex]
6. Central Angle in Radians:
The central angle in radians is found by multiplying this fraction by [tex]\(2 \pi\)[/tex] (since the full circle corresponds to [tex]\(2 \pi\)[/tex] radians):
[tex]\[ \text{Central Angle (radians)} = 2 \pi \times \frac{43}{60} = \frac{86 \pi}{60} \][/tex]
7. Simplify the Angle in Radians:
Simplify the fraction:
[tex]\[ \text{Central Angle (radians)} = \frac{43 \pi}{30} \approx 4.5 \text{ radians} \][/tex]
8. Convert Radians to Degrees:
To convert radians to degrees, use the conversion factor [tex]\( \frac{180}{\pi} \)[/tex]:
[tex]\[ \text{Central Angle (degrees)} = 4.5 \times \frac{180}{\pi} \approx 258.0^\circ \][/tex]
Thus, the measure of the associated central angle for the arc they traveled is:
[tex]\[ 258.0^\circ \][/tex]
The central angle measures [tex]\(258.0^\circ\)[/tex].
1. Diameter of the Ferris Wheel:
The diameter of the Ferris wheel is given as 40 feet.
2. Circumference Calculation:
The circumference of the Ferris wheel can be calculated using the formula [tex]\( C = \pi \times \text{diameter} \)[/tex].
[tex]\[ C = \pi \times 40 \][/tex]
3. Distance Traveled:
They traveled a distance of [tex]\( \frac{86}{3} \pi \)[/tex] feet.
4. Fraction of the Circumference Traveled:
To find the fraction of the circumference they traveled, divide the distance traveled by the circumference:
[tex]\[ \text{Fraction} = \frac{\text{Distance Traveled}}{\text{Circumference}} = \frac{\left(\frac{86}{3} \pi\right)}{\pi \times 40} \][/tex]
5. Simplify the Fraction:
When simplifying the fraction, the [tex]\(\pi\)[/tex] terms cancel out:
[tex]\[ \text{Fraction} = \frac{86}{120} = \frac{43}{60} \][/tex]
6. Central Angle in Radians:
The central angle in radians is found by multiplying this fraction by [tex]\(2 \pi\)[/tex] (since the full circle corresponds to [tex]\(2 \pi\)[/tex] radians):
[tex]\[ \text{Central Angle (radians)} = 2 \pi \times \frac{43}{60} = \frac{86 \pi}{60} \][/tex]
7. Simplify the Angle in Radians:
Simplify the fraction:
[tex]\[ \text{Central Angle (radians)} = \frac{43 \pi}{30} \approx 4.5 \text{ radians} \][/tex]
8. Convert Radians to Degrees:
To convert radians to degrees, use the conversion factor [tex]\( \frac{180}{\pi} \)[/tex]:
[tex]\[ \text{Central Angle (degrees)} = 4.5 \times \frac{180}{\pi} \approx 258.0^\circ \][/tex]
Thus, the measure of the associated central angle for the arc they traveled is:
[tex]\[ 258.0^\circ \][/tex]
The central angle measures [tex]\(258.0^\circ\)[/tex].
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