Question

Two gears are connected and are rotating simultaneously. The smaller gear has a radius of 4 inches, and the larger gear has a radius of 7 inches.
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Part 1: What is the angle measure, in degrees and rounded to the nearest tenth, through which the larger gear has rotated when the smaller gear has made one complete rotation?

Part 2: How many rotations will the smaller gear make during one complete rotation of the larger gear?
Show all work.

Answer 1

If anyone could possibly explain this step by step that would be amazing :)

Answer 2

\[2\pi*r\]
Do that for both the large and small circumference.

Answer 3

you will need the circumference formula later

but since all circles are similar the ratio of the radii will also be the ratio of circumference.

so this is how I thought about it...

for 4 rotations of the large gear you get 7 rotations of the small gear...

so then using that idea

1 rotation of the small gear will yield 4/7 of a rotation of the large gear..

so for part A find 4/7 of 360 degrees..

for part B use the reverse

4 large = 7 small so

1 large = ? small

hope it helps

Answer 4

oops missed the radian bit... finf 4/7 of 2pi keep it as an exact value

Answer 5

This problem boils down to the arc-length formula:
arc-length = \(r\theta\). where \(\theta\) is in radians.

Since the two gears wheels are in contact and engaged, the arc-length generated by rotation of each must equal, hence we equate:
\(r_1\theta_1=r_2\theta_2\) to solve similar problems.
Example:
r1=8"
r2=3"
When smaller gear makes one complete turn (\(\theta=2\pi\) ), then
8\(\theta_1\)=3*2pi
\(\theta_1\)=6pi/8=3pi/4

Answer 6

Thank you all so much for your help and time!!! I literally fell asleep waiting for help last night. Thx again :D