Question in finding final position in circle

Question

im having difficulty solving this problem right now, I tried couple of times but i couldnt figure it out, can you please help and explain?

thank you

Answer 1

For these questions, you'll want to use the formula for the arc length of a circle given by

$$L=\theta r$$

You want to find $L$ where $\theta$ is the angle measured from the starting position to the end position in radians (not degrees) and $r$ is the distance from the center of the circle (in this case, 27 cm).

You already know $r$, so each question is essentially asking you to find a new $\theta$.

Imagine the center of the fan is at the origin of the Cartesian plane.They give you the horizontal distance from the y-axis (i.e., vertical diameter), so you'll need to set up right triangles and use your trig functions to solve for the angle formed.

For part a), we know that the the right triangle formed has a side length 6 and hypotenuse 27, so $\arcsin(6/27) \approx .224$ radians. Add that to the $\frac{\pi}{2}$ radians formed by the bug traveling through the first quadrant and we get $\theta \approx 1.794$ radians.

\begin{align} L &\approx 27 \cdot 1.794 \\ & \approx 48.438 \, \textrm{cm} \end{align}

Repeat for the rest of the parts setting up a different triangle each time.