Question

Modeling Distance Traveled A car starts from point \(P\) at time \(t=0\) and travels at 45 mph. (a)Write an expression \(d(t)\) for the distance the car travels from \(P\) (b) Graph \(y=d(t) .\) (c) What is the slope of the graph in (b)? What does it have to do with the car? (d) Writing to Learn Create a scenario in which \(t\) could have negative values. (e) Writing to Learn Create a scenario in which the \(y\) -intercept of \(y=d(t)\) could be \(30 .\)

Short answer

The distance function is \(d(t) = 45t\), and the graph is a straight line from the origin with a slope of 45, representing that the car is traveling at a constant speed of 45 mph. Negative \(t\) values can be imagined if we started measuring after the car started. The \(y\)-intercept of \(30\) can be seen if the car had already traveled 30 miles when we have started tracking its motion.

Step-by-step solution

- Formulate the distance function

The car travels at a constant speed of 45 mph, so the distance can be modelled by a linear function, \(d(t) = 45t\), where \(t\) represents the time in hours. The distance, \(d(t)\), is the product of speed and time.

- Graph the function

To graph the function, take the horizontal axis as \(t\) and vertical axis as \(d\). As the function \(d(t) = 45t\) is a linear function, it will appear as a line passing through the origin and has a slope of 45. This line continues indefinitely representing the car keeps traveling with the same speed.

- Interpret the slope of the graph

The slope of the function \(d(t) = 45t\) is equal to the speed of the car, 45 mph. Slope represents the change in distance with respect to time, which is the definition of speed or velocity in Physics.

- Scenario for negative t values

Negative \(t\) values could be conceptualized if we assume the time we started measuring is actually some time after the car started from point \(P\). So, if the car started at 2 pm, and we started measuring at 3 pm, then 2 pm would correspond to \(t = -1\) hour.

- Scenario for non-zero y-intercept

y-intercept of \(y=d(t)\) could be \(30\) if the car had already travelled 30 miles when we started tracking its motion at \(t = 0\). In such a case, the function would look like \(d(t) = 45t + 30\).