Final answer:
The dashboard gauge to use when deciding to change gears is the tachometer, which measures engine RPM. The odometer measures total distance traveled, while the speedometer displays instantaneous speed, not velocity. Average speed is calculated by dividing total distance by trip time and is only the same as average velocity in straight-line round trips.
Step-by-step explanation:
The dashboard gauge that should be used to help decide when to change gears is b. The tachometer. A tachometer measures the engine's rotational speed in revolutions per minute (RPM). When the RPM reaches a certain level, that indicates it's time to change to a higher gear to maintain the engine's efficiency and prevent over-revving, which can cause damage. Conversely, when RPM drops too low, it's an indication that a downshift may be necessary to keep the engine running smoothly without stalling.
A car's odometer measures the total distance traveled over time, which is different from displacement. Displacement refers to the change in position from a starting point, in a straight line and can end up being less than the distance traveled if the path taken was not direct. On the other hand, a speedometer measures a car's instantaneous speed, not its velocity. Velocity would include both speed and direction, whereas speed is only concerned with how fast the vehicle is moving regardless of direction.
If you were to divide the total distance traveled (as determined by the odometer) by the time for the trip, you would be calculating the average speed, not the magnitude of the average velocity. These two quantities would be the same only if the car's travel was in a straight line and ended at the starting point, which rarely happens in normal driving conditions. Instantaneous speed is the speed of the car at a specific instant, which a speedometer displays, while instantaneous velocity includes the direction at that instant as well.