MATHEMATICS  CASE STUDY SOLUTION

Second 
Velocity 
0 
25 
1 
28 
2 
29 
3 
31 
4 
33 
5 
37 
6 
45 
7 
52 
8 
66 
9 
85 
10 
110 


The velocity of a car for
each second is shown in the table on the left. Estimate the total distance traveled by the car.
Let the distance that the car travels
to be d. Let the speed of the vehicle be v.
The velocity of the car in each second is shown in the graph.
Connect the points in the Velocity graph to form a
smooth curve. Recall that the velocity multiplied by the time
is the distance. In the diagram, the xaxis is time and yaxis
is velocity. Therefore, the area under the curve between 0 and
10 is the distance the car has traveled. 



Velocity 

Velocity Curve 



Estimated Distance Using Left Endpoints 

Assume that the velocity of the moving car
remains unchanged during each second, so the area under the curve is
the sum of the area in each second as shown on the left diagram. That
is
In this equation, the velocity for each second is listed
in the table and the time interval is 1 second. Substituting the speed and
the time interval into the above equation gives






Discussion

Estimated Distance Using Right Endpoints
Estimated Distance Using Middle Endpoints


This problem can also be solved using the
right endpoints as shown on the left graph. The estimated distance is
As
another formulation, this problem can be solved using the middle endpoints
shown on the left diagram. The solution can be determined using the simulation
in this section. 



Reduce Time Interval


The result of the estimated distance using the left and
right endpoints are large. The way to reduce the difference between them
is to
minimize the
time interval. For example, changing the time interval from 1 second to
0.5 second, as shown, can significantly reduce the error. 


