# Understanding the Problem We are given a variable-speed transmission system with pulleys whose diameters can change from 4 inches to 8 inches. We need to find the **range of the velocity ratio** as the pulley diameters vary within this range. --- # Key Concepts - **Velocity Ratio (VR):** It is the ratio of the velocity of the driver pulley (input) to the velocity of the driven pulley (output). - **Pulleys in a belt drive:** The velocity ratio depends on the diameters of the pulleys: \[ \text{VR} = \frac{\text{Diameter of driver pulley}}{\text{Diameter of driven pulley}} \] - **Variable pulleys:** Since the diameters can change, the velocity ratio varies accordingly. --- # Step-by-Step Solution ### Step 1: Identify the Range of Pulley Diameters - Minimum diameter: \( D_{min} = 4\, \text{in} \) - Maximum diameter: \( D_{max} = 8\, \text{in} \) ### Step 2: Understand the Possible Configurations The velocity ratio depends on which pulley is the driver and which is the driven. The pulleys can be either: - **Driver pulley:** The one connected to the power source, usually the smaller pulley for higher speed. - **Driven pulley:** The one transmitting power to the load, usually larger for torque. Since the pulleys can change diameters from 4 in to 8 in, the VR can be: - **Maximum:** When the driver pulley is the smallest (4 in) and the driven pulley is the largest (8 in) - **Minimum:** When the driver pulley is the largest (8 in) and the driven pulley is the smallest (4 in) ### Step 3: Calculate the Range of Velocity Ratios - **Maximum VR:** \[ \text{VR}_{max} = \frac{D_{driver, min}}{D_{driven, max}} = \frac{4}{8} = .5 \] - **Minimum VR:** \[ \text{VR}_{min} = \frac{D_{driver, max}}{D_{driven, min}} = \frac{8}{4} = 2 \] ### Step 4: Final Range of Velocity Ratio \[ \boxed{ \text{VR} \in [.5, 2] } \] This means: - The velocity ratio can be as low as **.5** when the driver pulley is small and the driven pulley is large. - The velocity ratio can be as high as **2** when the driver pulley is large and the driven pulley is small. --- # **Final Answer** **The range of the velocity ratio is from .5 to 2.**