Cubic Model Suppose a company is testing the flexibility of a new type of fiberglass. The company decides to test this flexibility by creating a diving board out of the fiberglass material. The bend deflection from the horizontal of a diving board when someone stands on it is a cubic function of the formLaTeX: y=ax^3+bx^2+cx+d, y = a x 3 + b x 2 + c x + d ,where y is the deflection of the board (in 1000ths of an inch) from the horizontal rest position, and x is the distance from the secured end of the board. Below is a table of distance and deflection measurements.

LaTeX: x\:\:xfeet LaTeX: y\:\:ydeflection
0 0
1 116
2 448
3 972
(a) Use a coefficient matrix and the supplied data to find the coefficients a, b, and c in this equation

LaTeX: y=ax^3+bx^2+cx+d,

and write the particular function that models this data. [NOTE: You only need a 3x3 matrix to find a, b, and c, since the first data point from the table, (0, 0) tells you that d must be equal to zero. Why is this true?]

(b) What is the multiplicity of 0 as a zero of the function?

(c) What is the deflection of this board at the tip, if it is 8 feet long?

(d) Some functions used to model data only work in limited intervals of the domain. Explore the graph of your particular function and identify an interval where the model breaks down as a reasonable model for describing the diving board’s deflection.