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.
[tex]\theta=\cos ^{-1}\frac{4.5}{10.3}=1.11865[/tex]So, the length of the arc =
[tex](2\pi-\theta)\cdot r=(2\pi-1.11865)\cdot10.3=53.194[/tex]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.
[tex]\theta=\sin ^{-1}\frac{4.7}{10.3}=0.4738[/tex]So, the length of the arc =
[tex](\frac{\pi}{2}+\theta)\cdot10.3=21.0593[/tex]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 =
[tex](\frac{3\pi}{2}-\theta)\cdot10.3=43.6575[/tex]
