Question

(01.07 mc) a driver takes a trip away from home. this person’s position during the five-hour drive is plotted on the graph below. the starting position (0, 0) is the driver’s home. identify the time period(s) during the trip when the car was traveling with the greatest average speed. a first hour b second hour c third hour d fourth

Explanation:

To determine the time period with the greatest average speed, we use the formula for average speed: \( \text{Average Speed} = \frac{\text{Change in Position}}{\text{Change in Time}} \). On a position - time graph, the slope of the line segment representing a time interval is equal to the average speed (since slope \( m=\frac{\Delta y}{\Delta x} \), and here \( \Delta y \) is the change in position and \( \Delta x \) is the change in time). A steeper slope means a greater average speed.

Step 1: Analyze the first hour

From \( t = 0 \) to \( t = 1 \) hour, the starting position is \( (0,0) \) and the ending position is \( (1, 40) \) (assuming the y - axis is position in miles). The change in position \( \Delta y=40 - 0 = 40 \) miles and the change in time \( \Delta x = 1-0=1 \) hour. The slope (average speed) is \( \frac{40}{1}=40 \) miles per hour.

Step 2: Analyze the second hour

From \( t = 1 \) to \( t = 2 \) hours, the position remains constant (the graph is horizontal). So the change in position \( \Delta y = 0 \). Using the average speed formula \( \text{Average Speed}=\frac{0}{1}=0 \) miles per hour.

Step 3: Analyze the third hour

From \( t = 2 \) to \( t = 3 \) hours, let's assume the starting position at \( t = 2 \) is 40 miles and at \( t = 3 \) is 60 miles. The change in position \( \Delta y=60 - 40 = 20 \) miles and the change in time \( \Delta x=3 - 2 = 1 \) hour. The average speed is \( \frac{20}{1}=20 \) miles per hour.

Step 4: Analyze the fourth hour

From \( t = 3 \) to \( t = 4 \) hours, if the starting position at \( t = 3 \) is 60 miles and at \( t = 4 \) is 80 miles. The change in position \( \Delta y=80 - 60 = 20 \) miles and the change in time \( \Delta x = 4-3=1 \) hour. The average speed is \( \frac{20}{1}=20 \) miles per hour.

Answer:

a. First hour