Question

two gears are adjusted so that the smaller gear drives the larger one, as shown in the figure. if the smaller gear rotates through an angle of 270°, through how many degrees will the larger gear rotate? the larger gear rotates through approximately ° (do not round until the final answer. then round to the nearest integer as needed )

Explanation:

Step1: Use the ratio of radii and angle - relationship

The arc - lengths of the two gears in contact are equal. Let the radius of the smaller gear be $r = 3.5$ cm and the radius of the larger gear be $R=7$ cm. Let the angle of rotation of the smaller gear be $\theta_1 = 270^{\circ}$ and the angle of rotation of the larger gear be $\theta_2$. The formula for arc - length is $s = r\theta$ (where $\theta$ is in radians). First, convert $\theta_1$ to radians: $\theta_1=270\times\frac{\pi}{180}=\frac{3\pi}{2}$ radians. Since $s_1 = s_2$, we have $r\theta_1=R\theta_2$.

Step2: Solve for $\theta_2$ in radians

We can rewrite the equation $r\theta_1 = R\theta_2$ as $\theta_2=\frac{r\theta_1}{R}$. Substituting $r = 3.5$, $R = 7$, and $\theta_1=\frac{3\pi}{2}$ into the equation, we get $\theta_2=\frac{3.5\times\frac{3\pi}{2}}{7}=\frac{3.5\times3\pi}{2\times7}=\frac{3\pi}{4}$ radians.

Step3: Convert $\theta_2$ to degrees

To convert $\theta_2$ from radians to degrees, use the conversion formula $\theta_{degrees}=\theta_{radians}\times\frac{180}{\pi}$. So, $\theta_2=\frac{3\pi}{4}\times\frac{180}{\pi}=135^{\circ}$.

Answer:

$135$