1. **Problem:** A cylindrical roller has a diameter of 48 inches and makes 0.7 revolutions per second. Find the angular speed in radians per second and the linear speed in inches per second. 2. **Formula and rules:** - Angular speed $\omega$ in radians per second is given by $\omega = 2\pi \times \text{revolutions per second}$. - Linear speed $v$ is related to angular speed by $v = r \times \omega$, where $r$ is the radius. 3. **Calculate angular speed:** $$\omega = 2\pi \times 0.7 = 1.4\pi \approx 4.398 \text{ rad/s}$$ 4. **Calculate radius:** $$r = \frac{\text{diameter}}{2} = \frac{48}{2} = 24 \text{ inches}$$ 5. **Calculate linear speed:** $$v = r \times \omega = 24 \times 1.4\pi = 33.6\pi \approx 105.6 \text{ in/s}$$ **Final answers:** - Angular speed $\approx 4.398$ rad/s - Linear speed $\approx 105.6$ in/s