When you run a race the person who crosses the finish line first is the winner. In this problem, the runner that reaches the 10 meter mark first wins the race. So, which runner who reaches the 10 meter mark first? To answer this question, we find out “when” each runner reaches the 10 meter mark.
We let Y equal the distance from the starting point and x equal the elapsed time. Then, we graph the functions for each runner,
Y1 = 2.4+0.75x
Y2 = .2x(x-5.1)
Y3 = .2x(x-5)(x-9.1),
Now, we want to find the time at which each runner reaches the 10 meter mark . To see this in the graph window, we draw the finish line. That is, we draw a horizontal line at Y = 10. Now, the time the runner takes to reach the 10 meter mark is the x coordinate of the point of intersection of the line Y= 10 and the graph of the runners function. Of course the Y coordinate is 10.
The “approximate” times are
for Y1, t = 10.133
for Y2, t = 10.067
forY3, t = 9.693
Don’t be misled when you first graph the three functions and they appear to intersect at the same point. If we repeatedly zoom in on the graphs around the point at which they appear to intersect, we eventually see the graphs separate and that they have no common point of intersection.
You can also solve algebraically by letting Y = 10, setting each of the runner’s functions to 10 and solving the equations:
2.4+0.75x = 10
.2x(x-5.1) = 10
.2x(x-5)(x-9.1) = 10.
But, the third equation is probably too difficult to solve algebraically. So we use a graphical or numerical method to find approximate answers. If you have any questions please respond.