Here is a simple, fun problem in mathematical modeling which can be analyzed on the TI-82 Graphics Calculator using parametric equations to represent graphically the foot marks of a walk.
Perhaps you've walked down a sidewalk avoiding, or stepping on, the cracks of the sidewalk. This problem presents a problem of that scenario.
Let's assume that the walkway is constructed in square meter blocks. Furthermore, assume that all the cracks are uniformly placed one meter apart and that we will neglect the measure of the width of the cracks. Also, the initial position of both feet is just outside the first crack.
Now, let's begin our walk by stepping first with the right foot. Your pace or step is a constant called P with a length of 76 centimeters, measured from the tip of the toe of one foot to the tip of the toe of the other foot. Also assume the shoe to have length F, which for this problem is 10 centimeters.
Given these conditions the question arises, "How many times, if at all, will either foot land on a crack?"
To attack this problem we suggest making a sketch and creating a table. This should lead you to see that the variables are: the number of steps, X, and the position of the feet from the start, in meters, which we will label Y. Using this information we've established the following linear equations: left foot represented by Y1 = P(X+1); and for the right foot, Y2 = PX. Also store 0.76 meters for the value of the pace P.
As you look at the graph, you can see that the function graphs of this system of equations do not adequately represent the position of your feet as you move down the path. Therefore, we suggest a parametric representation of these data.
In parametric form, let X1T equal distance in meters of the left foot from the start and X2T equal the distance of the right foot from the start. T will represent the number of steps taken. Using these conditions we define the equations as: X1T = P(T+1); Y1T = 5; X2T = PT; Y2T = 4. As a reminder, X1T represents the left foot and X2T the right. The values Y1T = 5, and Y2T = 4, are needed to separate the horizontal graphs of these data.
We suggest these window values: Tmin = 1, Tmax = 10, Tstep = 1, Xmin = 0, Xmax = 7.6 (Tmax * P), Xscl = 1, Ymin = 3, Ymax = 6, Yscl = 1. Remember that the stored value of P, the pace, was 0.76, and that the calculator needs to be in the "Dot" and "Simul" mode.
Looking at this graph and tracing, it is obvious that we need to distinguish between the left and right foot. To do this let each step of the right foot be given an odd number designation (the first step is taken by the right foot) and each step of the left foot represented by an even number. We can now model this problem in the parametric mode of the TI-82.
We offer these equations: X1T = (PT)/(fPart(T/2)=0); Y1T = 5; and X2T = (PT)/(fPart(T/2)0); Y2T = 4. Please note that the (T+1) term becomes just T for the left foot as a consequence of this counting system. Using the same window and by pressing GRAPH, the screen shows a graphical model of where your feet are as you walk down the path. Pressing TRACE highlights the step number and the position of each step along the path.
Now we need to consider the size of the shoe, or foot - in Arkansas, so that we may determine when you will land on a crack. Since we have decided that the pace is measured from the tip of the toe to the tip of a toe, if we find within a measure of 10 centimeters (F) back from the position of the tip of a toe that this interval covers a whole number measure, in meters, then you have stepped on a crack! Example: if the tip of the toe of your left foot is at 3.04 meters, the subtraction of an interval of 0.1 meters (length of shoe) encompasses a whole number measure of distance (3), which is where the cracks are located. Here is an algorithm that determines if any given step is on a crack. The form is 3.04 mod 100 < 0.1, or in general XT mod 100 < F. On the TI-82 you would key in the following, and turn off the X1T and X2T: X3T = X1T/(fPart(X1T)F ; Y3T = 5; X4T = X2T/(fPart(X2T)F; Y4T = 4. Graph this with the same window as before, with Xmax = Tmax * P, and you will see only points where your left or right foot land on a crack. To find the step number (T) and the distance from the start (X), press TRACE and manipulate the cursor until you hit a crack, which is indicated by a blinking cursor on the screen. Remember that the up and down keys will switch feet, and the right/left keys move you down the path. You will need to expand the T and X values in the window to see more steps.
Additional exploration with different sizes for your pace (P), and foot size (F) might reveal other patterns or foster additional questions.