Question

As a fan turns counterclockwise, a bug sits 10.3 inches 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. .... 4.5 inches to the right of the vertical diameter of the fan for the second time.____  inches   b. .... 4.7 inches to the left of the vertical diameter of the fan for the first time. _____ inches   c. .... 4.7 inches to the left of the vertical diameter of the fan for the second time.  ________inches

Answer 1

Given :

a bug sits 10.3 inches from the center of rotation on one of its blades.

So, the radius = r = 10.3

At first, The bug is at the 3 o'clock position on the fan when it begins to turn.

the length of the arc = θ * r

when , the bug sits 4.5 inches to the right of the vertical diameter of the fan for the second time.

`\theta=\cos ^(-1)(4.5)/(10.3)=1.11865`

So, the length of the arc =

`(2\pi-\theta)\cdot r=(2\pi-1.11865)\cdot10.3=53.194`

Note : for the second time the angle will be ( 2pi - θ )

b. when the bug 4.7 inches to the left of the vertical diameter of the fan for the first time.

`\theta=\sin ^(-1)(4.7)/(10.3)=0.4738`

So, the length of the arc =

`((\pi)/(2)+\theta)\cdot10.3=21.0593`

C. when the bug 4.7 inches to the left of the vertical diameter of the fan for the second time.

So, the length of the arc =

`((3\pi)/(2)-\theta)\cdot10.3=43.6575`