Professor Tapia examined the data and decided that the best way to answer this question was to determine where the funny car should be on the quarter-mile track when the dragster crosses the finish line. The distance between this point and the finish line is how far forward the funny car should be at the start of the "fair" race. Can you see why? Essentially, Professor Tapia shortened the race for the funny car by allowing it to only cover the portion of the quarter-mile track that it can complete in the same amount of time required for the dragster to cover the entire track. With this idea in mind, Professor Tapia was able to discover that the funny car should begin the race approximately 140 feet ahead of the dragster. [ XXX Some break here that would encourage readers to attempt to solve the problem rather than just reading on.]

Take a look at the graph of the data from the table for the funny car. Can you see where the funny car will be when the dragster crosses the finish line at 4.486 seconds? According to the graph, the funny car will be somewhere between 1000 and 1320 feet from the starting line. We cannot tell exactly where the funny car will be at 4.486 seconds because there is no data point on our graph at that time. But, you may be wondering, how did Professor Tapia know where the funny car would be when the dragster crosses the finish line?

Imagine that all of the data points were connnected somehow. Instead of having only six times during the race where we know how far along the funny car was, we would know where the funny car was at each point in time during the race. More importantly, we would know exactly where the funny car was at 4.486 seconds.

Figuring out how to connect the data points correctly involves a process called data fitting. By fitting the NHRA data to a curve Professor Tapia was able to determine where the funny car would be at 4.486 seconds, and to provide an answer to the queston posed by the NHRA.

Professor Tapia fit the data using a polynomial. The general equation for a polynomial is

f(x)= c_0 + c_1*x^1 + c_2*x^2 + c_3*x^3 + .... + c_{n-1}*x^{n-1} + c_n*x^n

where c_i for i= 1 ... n are the coefficients and n is the degree of the polynomial. The polynomial that fits the NHRA data gives the distance in feet, f(x), at any time during the race, x. That is,

f(0)= 0 f(0.885)= 60 f(2.277)= 330 f(3.266) = 660 f(4.091) = 1000 f(4.612) = 1254 f(4.788) = 1320

The value f(x) of this polynomial is known at six values of x. What is not known, however, is the value of the coefficients or the degree of the polynomial.

Professor Tapia decided to look for a polynomial with degree 6 because there are seven data points (providing seven equations), and a degree six polynomial has seven unknown coefficients. A degree 6 polynomial has the form

f(x)= c_0 + c_1*x^1 + c_2*x^2 + c_3*x^3 + c_4*x^4 + c_5*x^5 + c_6*x^6

As you may already know, having the same number of unknowns as equations allows us to create and, hopefully, solve a linear system.

To find the values of these coefficients, Professor Tapia created a system of linear equations by plugging each pair of data points from the NHRA into (\ref{deg6}). The system is as follows:

c_0 + 0*c_1 + 0*c_2 + 0*c_3 + 0*c_4 + 0*c_5 + 0*c_6 = 0 c_0 + 0.885*c_1 + 0.783*c_2 + 0.693*c_3 + 0.613*c_4 + 0.543*c_5 + 0.480*c_6 = 60 c_0 + 2.277*c_1 + 5.185*c_2 + 11.805*c_3 + 26.881*c_4 + 61.209*c_5 + 139.373*c_6 = 330 c_0 + 3.266*c_1 + 10.667*c_2 + 34.838*c_3 + 113.780*c_4 + 371.604*c_5 + 1213.660*c_6 = 660 c_0 + 4.091*c_1 + 16.736*c_2 + 68.468*c_3 + 280.103*c_4 + 1145.902*c_5 + 4687.884*c_6 = 1000 c_0 + 4.612*c_1 + 21.270*c_2 + 98.100*c_3 + 452.436*c_4 + 2086.635*c_5 + 9623.561*c_6 = 1254 c_0 + 4.788*c_1 + 22.925*c_2 + 109.765*c_3 + 525.553*c_4 + 2516.348*c_5 + 12048.274*c_6 = 1320

Solving this system gives us the values of the six coefficients of the polynomial. That is,

c_0 = 0 c_1 = 71.682 c_2 = -60.427 c_3 = 84.710 c_4 = -27.769 c_5 = 4.296 c_6 = -0.262

Therefore, the polynomial that interoplates the NHRA data is

f(x)= 71.682*x + -60.427*x^2 + 84.710*x^3 + -27.769*x^4 + 4.296*x^5 + -0.262*x^6.

The following figure contains a plot of this polynomial along with the NHRA data points.