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 13.3 inches. the bicycle will travel approximately □ in. (round to the nearest tenth.)

Explanation:

Step1: Find the ratio of the radii

The ratio of the radii of the two sprockets is $\frac{r_1}{r_2}=\frac{4.89}{1.05}$, where $r_1 = 4.89$ in is the radius of the larger sprocket and $r_2=1.05$ in is the radius of the smaller sprocket.

Step2: Determine the angle of rotation of the smaller sprocket

When the pedals (attached to the larger sprocket) are rotated through $\theta_1 = 180^{\circ}=\pi$ radians. Let $\theta_2$ be the angle of rotation of the smaller sprocket. Since the linear - distance traveled by the chain is the same for both sprockets, $r_1\theta_1=r_2\theta_2$. So, $\theta_2=\frac{r_1}{r_2}\theta_1=\frac{4.89}{1.05}\times\pi$ radians.

Step3: Calculate the distance the bicycle moves

The distance $d$ the bicycle moves is equal to the arc - length of the bicycle wheel. The formula for the arc - length of a circle is $s = r\theta$, where $r = 13.3$ inches is the radius of the bicycle wheel and $\theta=\theta_2$.
\[

$$\begin{align*} d&=13.3\times\frac{4.89}{1.05}\times\pi\\ &=13.3\times\frac{4.89\pi}{1.05}\\ &\approx13.3\times\frac{4.89\times 3.14}{1.05}\\ &=13.3\times\frac{15.3546}{1.05}\\ &=13.3\times14.6234286\\ &\approx194.5 \end{align*}$$

\]

Answer:

$194.5$