Answer:
Part 1: Angle = 154.3°
Part 2: About 2.33 Rotations
Step-by-step explanation:
Part 1:
The arc length of a circle is given by the formula [tex]s=r\theta[/tex]
Where
- s is the arc length
- r is the radius of the circle
- [tex]\theta[/tex] is the angle
**The angle are in radians and we will use radian**
When smaller gear has made 1 complete rotations ([tex]2\pi[/tex] radians), the arc length traveled is [tex]s=r\theta\\s=(3)(2\pi)\\s=6\pi[/tex]
Now, what angle is swept by the larger circle when it travels [tex]6\pi[/tex]? Put in [tex]s=6\pi[/tex] and r is 7 to find [tex]\theta[/tex]:
[tex]s=r\theta\\6\pi=(7)\theta\\\theta= \frac{6\pi}{7}[/tex]
** Since [tex]\pi[/tex] radians is 180°, we plug in 180 into [tex]\pi[/tex] and find the degree measure:
[tex]\frac{6\pi}{7}=\frac{6*180}{7}=154.3[/tex]
Angle = 154.3°
Part 2:
In one complete rotation of the larger gear, it travels a distance of:
[tex]s=r\theta\\s=(7)(2\pi)\\s=14\pi[/tex]
We know that the smaller circle makes 1 rotation and travels [tex]s=r\theta=(3)(2\pi)=6\pi[/tex], so in [tex]14\pi[/tex] distance, it makes:
Rotations = [tex]\frac{14\pi}{6\pi}=\frac{14}{6}=2.33[/tex]
About 2.33 Rotations
