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 300°, 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 the relationship between arc - lengths of two meshed gears

The arc - lengths along the circumferences of the two meshed gears are equal when one gear drives the other. The formula for the arc - length is $s = r\theta$, where $s$ is the arc - length, $r$ is the radius, and $\theta$ is the angle in radians. First, convert the angle of the smaller gear from degrees to radians. The angle of the smaller gear $\theta_1=300^{\circ}=300\times\frac{\pi}{180}=\frac{5\pi}{3}$ radians, and let the radius of the smaller gear $r_1 = 3.9$ cm and the radius of the larger gear $r_2=7.1$ cm. Let the angle of the larger gear be $\theta_2$ (in radians). Since $s_1 = s_2$, we have $r_1\theta_1=r_2\theta_2$.

Step2: Solve for $\theta_2$

We can rewrite the equation $r_1\theta_1=r_2\theta_2$ as $\theta_2=\frac{r_1\theta_1}{r_2}$. Substitute $r_1 = 3.9$, $\theta_1=\frac{5\pi}{3}$, and $r_2 = 7.1$ into the equation: $\theta_2=\frac{3.9\times\frac{5\pi}{3}}{7.1}=\frac{3.9\times5\pi}{3\times7.1}=\frac{19.5\pi}{21.3}$.

Step3: Convert $\theta_2$ from radians to degrees

To convert $\theta_2$ from radians to degrees, use the conversion factor $\theta_{degrees}=\theta_{radians}\times\frac{180}{\pi}$. So $\theta_{degrees}=\frac{19.5\pi}{21.3}\times\frac{180}{\pi}=\frac{19.5\times180}{21.3}$.

Step4: Calculate the value and round

$\frac{19.5\times180}{21.3}=\frac{3510}{21.3}\approx165$.

Answer:

165