Arc Length and Area of Sector│Please Help!

Question

Two connected gears are rotating. The smaller gear has a radius of 4in and the larger gear’s radius is 7in. What is the angle through which the larger gear has rotated when the smaller gear has rotated 120 degrees? Note that the gears have the same arc length.

1. Since you were given the info about the smaller gear, use that to find the arc length.
2. Then use the arc length you found and the given radius to find the missing angle for the larger gear.

Answer 1

Good question!

The formula for an arc is L = θ × r

L = length of the arc

θ = central angle of the arc

r = radius of the circle (or gear)

To solve your problem, the steps you would take look like this:

1. Solve for the L (arc length) of the smaller gear by substituting the information you know into the formula (θ = 120° or 2π/3; r = 4)
2. L = θ × r
3. L = 120 × 4
4. Since the arc lengths for both gears are the same, we can use the arc length that we found for the small gear and the formula to find the missing angle for the larger gear. (θ = unknown angle)
5. (small gear arc length) = θ × 7
6. (small gear arc length) / 7 = θ

I hope this helps! If you have any questions or need more detailed steps on how to solve the formula itself, please let me know, and I'd be happy to help! Good luck!