MHB Graphing Distance vs. Time for a Commuting Student's Forgotten Term Paper

[mathdad]

A student who commutes 27 miles to attend college remembers, after driving a few minutes, that a term paper that is due has been forgotten. Driving faster than usual, the student returns home, picks up the paper, and once again starts toward school. Sketch a possible graph of the student's distance from home as a function of time. What are the steps to answer this question?

 

[MarkFL]

What have you got so far?

 

[mathdad]

What have you got so far?

Mark,

I am on the 7 train in Queens, NYC heading back home to the Bronx. When I get home, I will post my understanding of this question, which will be wrong for sure. In fact, my work shown will not make sense and may cause some of you to ask: WHY DOESN'T THIS PRECALCULUS GUY UNDERSTAND WHAT TO DO AS A FIRST COUPLE OF STEPS?

 

[Joppy]

WHY DOESN'T THIS PRECALCULUS GUY UNDERSTAND WHAT TO DO AS A FIRST COUPLE OF STEPS?

No such thoughts will (or should) ever be conceived by a good teacher (as are those members here!). Any of your thoughts and working help others in identifying your particular misunderstanding. And if that means making several posts, progressively stepping through the problem, so be it.

 

[mathdad]

Let the x-axis = time of travel in hours

Let the y-axis = miles traveled

I think we are talking about a straight line or linear function graph.

The question does not state the amount of minutes the student drove before realizing that he forgot his term paper nor does it mentions the length he drove.

On the graph, A is the time in minutes he drove and B the length. See picture. Is this correct?

View attachment 7868

 

[MarkFL]

Looks good so far...now what's going to happen when the student heads back home at a faster speed?

 

[mathdad]

Looks good so far...now what's going to happen when the student heads back home at a faster speed?

The graph changes direction? Must I draw another arrow pointing downward?

 

[MarkFL]

The graph changes direction? Must I draw another arrow pointing downward?

Suppose the vertical axis is $d$ and the horizontal $t$, to represent distance and time, respectively. What does a point $(t,d)$ represent?

 

[mathdad]

Suppose the vertical axis is $d$ and the horizontal $t$, to represent distance and time, respectively. What does a point $(t,d)$ represent?

The point (t, d) represents the student's distance from home.

 

[MarkFL]

The point (t, d) represents the student's distance from home.

The vertical distance of the point above the $t$-axis represents this distance, while the horizontal distance of the point to the $d$-axis to its left represents the time.

When the student returns toward home, what should be happening to the coordinates of the points on the graph?

 

[mathdad]

The vertical distance of the point above the $t$-axis represents this distance, while the horizontal distance of the point to the $d$-axis to its left represents the time.

When the student returns toward home, what should be happening to the coordinates of the points on the graph?

The coordinates of the point (t, d) should be increasing or decreasing.

 

[MarkFL]

The coordinates of the point (t, d) should be increasing or decreasing.

As the student returns home, time is increasing (moving forward), but the distance from home is decreasing and so the slope of the graph will be negative. How will the magnitude of the slope compare to when the student was moving away from hom before turning back?

 

[mathdad]

As the student returns home, time is increasing (moving forward), but the distance from home is decreasing and so the slope of the graph will be negative. How will the magnitude of the slope compare to when the student was moving away from hom before turning back?

I do not understand? Magnitude of the slope?

 

[MarkFL]

I do not understand? Magnitude of the slope?

How steep will it be, in comparison to the first part of the trip?

 

[mathdad]

By "steep" you are talking about slope but I don't get it.

Origin = (0,0)

Our point is (t, d)

Can you steepness be found using these 2 points?

m = (t - 0)/(d - 0)

m = t/d

Does the value of m here satisfy the "steep" part of your question?

 

[MarkFL]

By "steep" you are talking about slope but I don't get it.

Origin = (0,0)

Our point is (t, d)

Can you steepness be found using these 2 points?

m = (t - 0)/(d - 0)

m = t/d

Does the value of m here satisfy the "steep" part of your question?

Excellent! Now, suppose the student travels back home twice as fast as he left...what will the slope of the second part of the trip be?

 

[mathdad]

The slope of the second part of his trip is 2(t/d).

 

[MarkFL]

The slope of the second part of his trip is 2(t/d).

Let's check:

$$m=\frac{\Delta d}{\Delta t}=\frac{0-d}{\frac{3}{2}t-t}=-\frac{d}{\frac{1}{2}t}=-2\frac{d}{t}$$

I didn't notice earlier that you mixed up $d$ and $t$.

 

[mathdad]

The answer is -2(d/t) not 2(d/t). Why a negative slope?

 

[MarkFL]

The answer is -2(d/t) not 2(d/t). Why a negative slope?

Shouldn't we expect a negative slope since the $d$ coordinate will be moving towards the $t$-axis from some positive value, towards where $d=0$?

 

[mathdad]

And so, the final answer to this question is?

 

[MarkFL]

It could look something like:

View attachment 7892

 

[mathdad]

It could look something like:

You are truly gifted. Thanks.