two gears are adjusted so that the smaller gear drives the larger one, as shown in the figure. if the…

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 )

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 )

Answer

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:

$135$