Refer to the data in Exercise 19 in Section 6.6, concerning the takeoff performance of an airplane. Suppose the possible measurement errors become greater as the speed of the airplane increases, and let W be the diagonal weighting matrix whose diagonal entries are 1, 1, 1, .9, .9, .8, .7, .6, .5, .4, .3, .2, and.1. Find the cubic curve that fits the data with minimum weighted least-squares error, and use it to estimate the velocity of the plane when t = 4:5 seconds.

Data From Exercise 19 in Section 6.6

To measure the takeoff performance of an airplane, the horizontal position of the plane was measured every second, from t = 0 to t = 12. The positions (in feet) were: 0, 8.8, 29.9, 62.0, 104.7,159.1,222.0,294.5, 380.4,471.1, 571.7,686.8, and 809.2.