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: Recall arc - length formula

The arc - length formula is $s = r\theta$, where $s$ is the arc - length, $r$ is the radius, and $\theta$ is the angle in radians. Since the arc - lengths of the two gears in contact are equal when the smaller gear drives the larger one. First, convert the angle of the smaller gear from degrees to radians. $\theta_1=270^{\circ}=270\times\frac{\pi}{180}=\frac{3\pi}{2}$ radians, $r_1 = 3.5$ cm and $r_2=7.3$ cm.

Step2: Set arc - lengths equal

Let $\theta_1$ be the angle of the smaller gear and $\theta_2$ be the angle of the larger gear. Since $s_1 = s_2$, we have $r_1\theta_1=r_2\theta_2$. Substitute the known values: $3.5\times\frac{3\pi}{2}=7.3\times\theta_2$.

Step3: Solve for $\theta_2$

$\theta_2=\frac{3.5\times\frac{3\pi}{2}}{7.3}=\frac{3.5\times3\pi}{2\times7.3}=\frac{10.5\pi}{14.6}$ radians.

Step4: 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 $\theta_2=\frac{10.5\pi}{14.6}\times\frac{180}{\pi}=\frac{10.5\times180}{14.6}=\frac{1890}{14.6}\approx129.45$.

Answer:

$129$