Problem $9\mathrm{.}6$

An engineer applies a tension force to a piece of steel as shown in the figure below.

The engineer is able to measure the change in length of the beam due to the applied force with a high precision laser. The engineer measured the following data:

Knowing that the data were collected with high precision, you want to use spline interpolation to predict the change in the length of the beam at other applied loads. Use spline interpolation to fit $3$ cubic polynomials to the $4$ data points and then predict the change in the length of the beam due to any $3$ applied loads of your choice $($besides the test case shown below$).$ On a single plot, show the experimental data points with corresponding to each interpolating polynomial. On a diary file, predict the change of length for the beam for an applied load of $5$ KN $.$