Julieta has been in the car with her family on a road trip for the past three hours. For the first hour, they went about 55 mph. Then, the speed limit changed and they started going about 70 mph. They are still going 70 mph.

Question

A. Create a graph that models the distance Julieta’s family has traveled over time during their road trip.

B. What does the slope of your graph at any given time represent? Explain.

C. Create a piecewise function that models the distance Julieta's family has
traveled over time.

D. Two and a half hours into the road trip, how far had Julieta’s family traveled? Give your answer to a reasonable level of accuracy. Explain why your response cannot be perfectly accurate.

E. Write your own question that asks a student to interpret a different feature of your graph/function in terms of the given context. Then, provide the solution for your question.

Answer 1

A. Graph of Distance vs. Time

To model the distance Julieta's family has traveled over time, we can create a piecewise graph. The x-axis represents time (in hours), and the y-axis represents the distance traveled (in miles).

1. In the first hour (from 0 to 1 hour), they travel at 55 mph.

   • Distance = Speed × Time = 55 mph × 1 hour = 55 miles.

2. For the next 2 hours (from 1 to 3 hours), they travel at 70 mph.

   • Distance during this segment = Speed × Time = 70 mph × 2 hours = 140 miles.

The total distances at different times are:

• At 0 hours: 0 miles
• At 1 hour: 55 miles
• At 3 hours: 55 miles + 140 miles = 195 miles

To graph this:

• From (0, 0) to (1, 55): This section has a slope of 55 (rise/run).
• From (1, 55) to (3, 195): This section has a slope of 70.

B. Slope Interpretation

The slope of the graph at any given time represents the speed of Julieta's family. A steeper slope indicates a higher speed. For the first hour, the slope is 55, representing 55 mph – the speed at which they were traveling. For the next two hours, the slope is 70, reflecting their increased speed of 70 mph.

C. Piecewise Function

The distance \( d(t) \) as a piecewise function can be represented as follows:

\[ d(t) = \begin{cases} 55t & \text{for } 0 \leq t \leq 1 \ 55 + 70(t - 1) & \text{for } 1 < t \leq 3 \end{cases} \]

In this function:

• In the first case (for \( 0 \leq t \leq 1 \)), the distance is the product of speed and time.
• In the second case (for \( 1 < t \leq 3 \)), we add the distance traveled in the first hour (55 miles) to the distance traveled at 70 mph for the remaining time.

D. Distance Traveled at 2.5 Hours

To find the distance traveled at 2.5 hours, we can use the piecewise function. Since 2.5 hours is in the second segment of the piecewise function:

\[ d(2.5) = 55 + 70(2.5 - 1) \] \[ = 55 + 70(1.5) \] \[ = 55 + 105 = 160 \text{ miles} \]

Thus, Julieta’s family has traveled 160 miles at 2.5 hours.

The response cannot be perfectly accurate due to several factors, such as variability in driving speed due to traffic, road conditions, or any stops they might have made along the journey.

E. Question and Solution

Question: What would be the distance traveled after 1.5 hours into the road trip, and how does this reflect the change in speed?

Solution: To find the distance at 1.5 hours, we must use the second piece of the piecewise function since 1.5 is greater than 1.

\[ d(1.5) = 55 + 70(1.5 - 1) \] \[ = 55 + 70(0.5) \] \[ = 55 + 35 = 90 \text{ miles} \]

At 1.5 hours, Julieta’s family has traveled 90 miles. This reflects the change in speed from 55 mph to 70 mph after the first hour, showing that their distance traveled increased more rapidly after the speed limit changed and they increased their traveling speed.