Circular Fan Travel Distance
A circular ceiling fan rotates at a constant speed of $ 80 $ revolutions per minute. A point halfway between the center of the fan and the outer edge of the fan travels $ 97968 $ inches in $ 15 $ minutes. How far (in inches) does a point on the outer edge of the fan travel in $ 30 $ minutes?
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Solution
Because the fan rotates at a constant speed, by doubling the time from 15 minutes to 30 minutes, points on the fan travel twice as far. Furthermore, in each rotation, the point on the outer edge of the fan travels twice as far a point halfway between the center of the fan and the outer edge. Therefore in 30 minutes a point on the outer edge of the fan travels $ 4 \times 97968 = \boxed{391872} $ inches.
$ \textbf{Alternate solution} $:
In 15 minutes, the fan makes $ 80\cdot15=1200 $ revolutions. That means in each revolution, the halfway point travels $ 97968/1200 $ inches. This is equal to the circumference of the circle on which the halfway point travels. Since circumference is equal to $ 2\pi r $, the radius $ r $ is equal to $ 97968/(1200\cdot2\cdot\pi)=97968/(2400\pi) $ inches. The radius of the circle on which the outer point travels is double the radius we found, or $ 97968/(1200\pi) $ inches, so the circumference is $ 2\pi\cdot97968/(1200\pi)=97968/600 $ inches. In 30 minutes, the outer point travels $ 2\cdot1200=2400 $ revolutions (there are 1200 revolutions in 15 minutes) around this circumference, so the point travels a total distance of $ 97968/600\cdot2400=\boxed{391872} $ inches.
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