Question

Problem 4 (hand-calculation): Consider the constant-pressure specific heat of air at high temperature presented in ta- ble 4, where T is the temperature and Cp is the specific heat. Determine a least squares quadratic polynomial approximation for this set of data. The quadratic polynomial has the following form: Cp = a + bT+cT. where the coefficients a, b and c are to be determined using the least squares method. Hint Follow the derivation of linear regression discussed in class. You can use MATLAB to solve a linear system of equations for the coefficients Temperature, T (K) 1000 1500 200025003000 Specific heat, Cp (k.J/kg K) 1.1410 12095 1.2520 1.2782 1.2955 Table 4: Property of air: Variation of specific heat with temperature. Take a photo of your work. Include all pages in a single photo named problem4.jpg. Set the following in your homework script: figure(4); İmshow/inread('problem4.jpg')); p4 'See figure 4'.

Answer 1

I am using R for calculating a,b,c

The codes and output are attached.

a=1.235240

b=0.119439

c=-0.031724

R RGui (32-bit) File Edit Packages Windows Help R R Console T-c (1000,1500, 2000,2500, 3000) >Cp c (1.141,1.2095,1.252,1.2782,1.2955) summary (1m (Cp poly (T,2))) Call: 1m (formula - Cppoly (T, 2)) Residuals: 1 2 3 4 -0.0017429 0.0035514 -0.0001971 -0.0032886 0.0016771 Coefficients: Estimate Std. Error t value Pr (>tI) (Intercept) 1.235240 0.001712 721.431 1.92e-06 poly (T, 2)1 0.119439 0.003829 31.196 0.00103 poly (1,2)2-0.031724 0.003829-8.286 0.01425 Signif. codes: 0 0.001 '**' 0.01 '*0.05 '.' 0.1''1 Residual standard error 0.003829 on 2 degrees of freedom Multiple R-squared: 0.9981, F-statistic: 520.9 on 2 and 2 DE, p-value: 0.001916 Adjusted R-squared: 0.9962