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?

Show all work.

Answer 1

Circumference of a circle is C = 2πr.

part 1)

so the radius of the smaller gear is 3 inches, so after a full rotation, namely a revolution, the smaller gear has covered an arc of 2π(3), or 6π.

what angle has the larger gear of 7 inches in radius, has it covered with an arc of 6π?

$\bf \textit{arc's length}\\\\ s=r\theta ~~ \begin{cases} r=radius\\ \theta =angle~in\\ \qquad radians\\ ------\\ s=6\pi \\ r=7 \end{cases}\implies 6\pi =7\theta \implies \cfrac{6\pi }{7}=\theta$



part 2)

since the larger gear has a radius of 7, in a revolution it will have an arc of 2π(7), or 14π.

now, we know the smaller gear does a 6π arc in a revolution, how many revolutions will it do in an arc of 14π?

$\bf \begin{array}{ccll} arc&revolution\\ \text{\textemdash\textemdash\textemdash}&\text{\textemdash\textemdash\textemdash}\\ 6\pi &1\\ 14\pi &x \end{array}\implies \cfrac{6\pi }{14\pi }=\cfrac{1}{x}\implies \cfrac{3}{7}=\cfrac{1}{x}\implies x=\cfrac{7}{3}$