INSTRUCTIONS. Include all intermediate steps in your calculations and ensure that your final answers are clearly

presented.

Problem #$1.(80$ points$).$ An experiment was conducted to determine the percentage of elongation $)$ of a

material as a function of temperature $(T)$ in $C.$ The data from the experiment is provided in the table below, along

with a plot of $\%$ elongation vs$.$ temperature shown in Figure #$1.$

Figure #$1.\%-$Elongation of a Material $(\%)$ vs$.$ Temperature $(T).$

Question $1.(40$ points$).$ From the plot in Figure #$1,$ it is evident that selecting appropriate data points allows for

approximating the percentage elongation of the material at a given temperature using simple linear interpolation.

Based on this observation, answer the following:$\text{'}$

$(20$ points$).$ Find the slope and intercept of the linear polynomial that approximates the relationship

between temperature and elongation for the relevant data points. Clearly show your selection of data points

from the table.

$(20$ points$).$ Using the slope and intercept in item $(1),$ calculate the percent elongation for a temperature

of $400C.$ Provide detailed calculations and explanations for finding the slope, intercept, and the

interpolated value.

Question $2.(25$ points$).$ The technician now believes that the data collected during the experiment may be "noisy."

As a result, she opts to use a "best$-$fit" solution to better approximate the relationship between elongation and

temperature. For this case, complete the following tasks:

$(10$ points$).$ Use the minimum$-$squared error $($MSE$)$ method to find the coefficients of the best$-$fit linear

polynomial $P_1(x)=a_0+a_{1}x.$

$(10$ points$).$ Using the best$-$fit polynomial and all the data points in the dataset, calculate the percentage

of elongation at a temperature of $400C.$ Provide a table with all the necessary values used to compute

the matrix of coefficients and to solve the simultaneous linear equations.

$(5$ points$).$ Calculate the percentage error if the percentage of elongation obtained from the best$-$fit method

is considered closer to the theoretical solution.

Question $3$: $(10$ points$)$ Would you consider using a non$-$linear interpolation polynomial to solve this problem

instead? Provide a detailed explanation of your reasoning, including the potential benefits or drawbacks of such an

approach.