Question

the graph in figure 1 shows time (minutes) on the x - axis and distance (miles) on the y - axis for cars traveling in the same direction along the freeway. the graph for car a is a straight line. the graph for car b is a parabola because it is a quadratic function.

1. which car has the cruise control on (is maintaining at the same speed)? how do you know?
2. which car is accelerating? how do you know?
3. identify the interval in figure 1 where car a has gone farther than car b.
4. the graph of the speed of a third car, car c, which has an exponential relationship is now shown in the graph (see figure 2). all 3 cars have the same destination.

a. if the destination corresponds with a distance of 12 miles from the origin, which car do you predict will arrive first? justify your answer.
b. if the three cars passed the starting point at the same time, and were racing, would there ever be another time that they would be tied? explain.
c. describe the race for these three cars.

1. if the cars are able to proceed beyond a time of 13 minutes, according to the type of function they are being described by, will the lead ever change again? explain.
2. on a graph that shows distance versus time, what do you look at in order to find speed?

Explanation:

Step1: Analyze speed - constant or changing

For a distance - time graph, a straight - line graph represents a constant speed because the slope (rate of change of distance with respect to time) is constant. A non - straight line (in this case, a quadratic function for Car B) represents a changing speed.
Car A has a straight - line distance - time graph. The slope of the distance - time graph gives the speed. Since the slope of Car A's graph is constant, Car A has the cruise control on.

Step2: Identify accelerating car

Car B has a non - straight (quadratic) distance - time graph. The slope of its graph is changing over time. An increasing slope indicates an increasing speed, which means the car is accelerating. So Car B is accelerating.

Step3: Find interval where Car A is ahead

By looking at the distance - time graph of Figure 1, we can see that Car A has gone farther than Car B in the interval \(0\leq t<5\) minutes.

Step4: Predict arrival time for destination

For Car A, it has a constant speed. For Car B, it is accelerating. For Car C, it has an exponential relationship. As the distance to the destination is 12 miles, Car C will arrive first. In an exponential function, as time progresses, the function grows at an increasingly faster rate. So, even though it may start slow, it will over - take the other cars.

Step5: Check for tie - ups

If the three cars start at the same time, they will be tied at the starting point (time \(t = 0\)). After that, since the functions representing their motion (linear for Car A, quadratic for Car B, and exponential for Car C) have different growth rates, they will not be tied again.

Step6: Describe the race

At the start, all cars are at the same point. Initially, Car A is ahead of Car B. Car C starts slow. As time progresses, Car B starts catching up with Car A due to its acceleration. Eventually, Car C, with its exponential growth, will over - take both Car A and Car B.

Step7: Analyze lead change after 13 minutes

Car C has an exponential function. Car A has a linear function and Car B has a quadratic function. As time goes on, the exponential function of Car C will grow much faster than the linear and quadratic functions. So, if the cars can proceed beyond 13 minutes, the lead will change again, with Car C taking the lead.

Step8: Determine speed from distance - time graph

On a distance - time graph, the speed is given by the slope of the graph at a particular point. If the graph is a straight line, the average speed is the slope of the line. If it is a curved line, the instantaneous speed at a point is the slope of the tangent line to the curve at that point.

Answer:

1. Car A has the cruise control on. We know this because its distance - time graph is a straight line, indicating a constant speed.
2. Car B is accelerating. We know this because its distance - time graph is a non - straight (quadratic) line, meaning its speed is changing (increasing).
3. The interval where car A has gone farther than car B is \(0\leq t<5\) minutes.
4. a. Car C will arrive first. Its exponential relationship will cause it to grow at an increasingly faster rate and over - take the other cars.

b. No, they will not be tied again after the starting point because their functions (linear, quadratic, exponential) have different growth rates.
c. At first, Car A is ahead of Car B. Car C starts slow. Then Car B starts catching up with Car A due to acceleration, and eventually Car C over - takes both.

1. Yes, the lead will change again. Car C's exponential function will grow faster than the linear and quadratic functions of Car A and Car B.
2. The slope of the distance - time graph gives the speed. For a straight - line graph, it's the slope of the line; for a curved graph, it's the slope of the tangent line at a point.