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

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 they mesh, we have $s_1=s_2$. Let $r_1 = 3.5$ cm be the radius of the smaller gear, $\theta_1$ be the angle of rotation of the smaller gear, $r_2=7.1$ cm be the radius of the larger gear, and $\theta_2$ be the angle of rotation of the larger gear. First, convert $\theta_1 = 270^{\circ}$ to radians. We know that $1^{\circ}=\frac{\pi}{180}$ radians, so $\theta_1=270\times\frac{\pi}{180}=\frac{3\pi}{2}$ radians.

Step2: Set up the arc - length equation

Since $s_1 = s_2$, we have $r_1\theta_1=r_2\theta_2$. Substitute $r_1 = 3.5$, $\theta_1=\frac{3\pi}{2}$, and $r_2 = 7.1$ into the equation: $3.5\times\frac{3\pi}{2}=7.1\times\theta_2$.

Step3: Solve for $\theta_2$

First, simplify the left - hand side of the equation: $3.5\times\frac{3\pi}{2}=\frac{10.5\pi}{2}$. Then, solve for $\theta_2$: $\theta_2=\frac{3.5\times\frac{3\pi}{2}}{7.1}=\frac{10.5\pi}{2\times7.1}=\frac{10.5\pi}{14.2}$. Convert $\theta_2$ back to degrees. We know that $\theta$ (in degrees) $=\theta$ (in radians) $\times\frac{180}{\pi}$. So $\theta_2=\frac{10.5\pi}{14.2}\times\frac{180}{\pi}=\frac{10.5\times180}{14.2}=\frac{1890}{14.2}\approx133^{\circ}$.

Answer:

$133$