Question

Two gears are connected and are rotating simultaneously. The smaller gear has a radius of 3 inches, and the larger gear has a radius of 7 inches.

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?

Answer 1

Part 1: Angle = 154.3°

Part 2: About 2.33 Rotations

Explanation:

Part 1:

The arc length of a circle is given by the formula
`s=r\theta`

Where

• s is the arc length
• r is the radius of the circle
• 
  `\theta` is the angle

**The angle are in radians and we will use radian**

When smaller gear has made 1 complete rotations (
`2\pi` radians), the arc length traveled is
`s=r\theta\\s=(3)(2\pi)\\s=6\pi`

Now, what angle is swept by the larger circle when it travels
`6\pi`? Put in
`s=6\pi` and r is 7 to find
`\theta`:

`s=r\theta\\6\pi=(7)\theta\\\theta= (6\pi)/(7)`

** Since
`\pi` radians is 180°, we plug in 180 into
`\pi` and find the degree measure:

`(6\pi)/(7)=(6*180)/(7)=154.3`

Angle = 154.3°

Part 2:

In one complete rotation of the larger gear, it travels a distance of:

`s=r\theta\\s=(7)(2\pi)\\s=14\pi`

We know that the smaller circle makes 1 rotation and travels
`s=r\theta=(3)(2\pi)=6\pi`, so in
`14\pi` distance, it makes:

Rotations =
`(14\pi)/(6\pi)=(14)/(6)=2.33`

About 2.33 Rotations