Question

the figure shows the chain drive of a bicycle. how far will the bicycle move if the pedals are rotated through 180°? assume the radius of the bicycle wheel is 12.3 inches. the bicycle will travel approximately in. (round to the nearest tenth.)

Explanation:

Step1: Find the ratio of the radii of the two sprockets

The ratio of the radii of the larger sprocket (connected to pedals) to the smaller sprocket (connected to the rear - wheel) is $\frac{4.91}{1.24}$.

Step2: Determine the angle of rotation of the rear - wheel

When the pedals are rotated through $180^{\circ}=\pi$ radians, the rear - wheel rotates through an angle $\theta$ such that the arc - lengths of the two sprockets are equal. Let the angle of rotation of the pedals be $\theta_1$ and the angle of rotation of the rear - wheel be $\theta_2$. Since the arc - length $s = r\theta$ and $s_1=s_2$, we have $r_1\theta_1=r_2\theta_2$. So, $\theta_2=\frac{r_1}{r_2}\theta_1$. Substituting $r_1 = 4.91$, $r_2 = 1.24$, and $\theta_1=\pi$ radians, we get $\theta_2=\frac{4.91}{1.24}\pi$ radians.

Step3: Calculate the distance the bicycle moves

The distance $d$ the bicycle moves is equal to the arc - length of the rear - wheel. The formula for the arc - length of a circle is $s = r\theta$, where $r$ is the radius of the rear - wheel and $\theta$ is the angle of rotation of the rear - wheel in radians. Given $r = 12.3$ inches and $\theta=\frac{4.91}{1.24}\pi$ radians, we have $d=12.3\times\frac{4.91}{1.24}\pi$.
\[

$$\begin{align*} d&=12.3\times\frac{4.91}{1.24}\times3.14\\ &=12.3\times\frac{15.4174}{1.24}\\ &=12.3\times12.4334\\ &\approx152.9 \end{align*}$$

\]

Answer:

$152.9$