In the diagram, wheel $B$ touches wheel $A$ and wheel $C$. Wheel $A$ has radius $35 \mathrm{~cm}$, wheel $B$ has radius $20 \mathrm{~cm}$, and wheel $C$ has radius $8 \mathrm{~cm}$. If wheel $A$ rotates, it causes wheel $B$ to rotate without slipping, which causes wheel $C$ to rotate without slipping. When wheel $A$ rotates through an angle of $72^{\circ}$, what is the measure of the angle through which wheel $C$ rotates?

![](https://cdn.mathpix.com/cropped/2024_04_30_d1e45f8438f865d01c92g-1.jpg?height=369&width=350&top_left_y=1927&top_left_x=1451)
Since the three wheels touch and rotate without slipping, then the arc lengths through which each rotates will all be equal.

Since wheel $A$ has radius $35 \mathrm{~cm}$ and rotates through an angle of $72^{\circ}$, then the arc length through which it rotates is $\frac{72^{\circ}}{360^{\circ}}(2 \pi(35 \mathrm{~cm}))=14 \pi \mathrm{cm}$. (This equals the fraction of the entire circumference as determined by the fraction of the entire central angle.)

If wheel $C$ rotates through an angle of $\theta$, then we must have $\frac{\theta}{360^{\circ}}(2 \pi(8 \mathrm{~cm}))=14 \pi \mathrm{cm}$.

Simplifying, we obtain $\frac{\theta}{360^{\circ}}=\frac{14}{16}$ and so $\theta=360^{\circ} \cdot \frac{7}{8}=315^{\circ}$.

Therefore, wheel $C$ rotates through an angle of $315^{\circ}$.

ANSWER: $315^{\circ}$