As a fan turns counterclockwise, a bug sits 1.8 feet from the center of rotation

Question

As a fan turns counterclockwise, a bug sits 1.8 feet from the center of rotation on one of its blades. The bug is at the 3 o'clock position on the fan when it begins to turn.

If the fan makes less than one full rotation, determine the distance the bug has traveled along its arc from the 3 o'clock position when its final position is...

a) ... 1 ft above the horizontal diameter of the fan for the second time. ________ ft

b) ... 0.5 ft below the horizontal diameter of the fan for the first time. _________ft

c) ... 1.3 ft below the horizontal diameter of the fan for the second time._________ft

Explanation / Answer

In this Example, we will draw a circle with a 1.8 ft radius. Then divide it into quadrants like the unit circle.

a) We want to know the second time the bug hits a certain point above the horizontal, so the point that we're looking for is in the second quadrant. If you draw a right triangle, we can use it to find the reference angle. Find the inverse sine of 1/1.8 (opposite/hypotenuse). It is equal to approximately 33.75 degrees. When we subtract that from 180, we get the number of degrees the bug has traveled so far which is 146.25 degrees. Now we need to find the arc length. Use the circumference formula (2 times pi times radius) and then multiply by 146.25/360. Answer is 4.59 feet

b) Same process, but now we are in the 3rd quadrant. Use inverse sine of 0.5/1.8 to find the reference angle 22.88 and add that to 180 to get the bug's angle 202.88. Find arc length ( 2 * pi * 1.8 * 202.88 / 360 = approximately 0.50 ft.)

c) Same process, 4th quadrant. sin-1(1.5/1.8) = 56.44 360- 56.44 = 303.56

arc length = 2 * pi * 1.8 * 303.56 / 360 = approximately 9.54 ft.

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