 # A small grocer finds that the monthly sales y (in $) can be approximated as a function of the amount spent advertising on the radio x_1 (in$) and the Wierzycaz 2020-11-06 Answered
A small grocer finds that the monthly sales y (in $) can be approximated as a function of the amount spent advertising on the radio ${x}_{1}$ (in$) and the amount spent advertising in the newspaper ${x}_{2}$ (in $) according to $y=a{x}_{1}+b{x}_{2}+c$ The table gives the amounts spent in advertising and the corresponding monthly sales for 3 months. a) Use the data to write a system of linear equations to solve for a, b, and c. b) Use a graphing utility to find the reduced row-echelon form of the augmented matrix. c) Write the model $y=a{x}_{1}+b{x}_{2}+c$ d) Predict the monthly sales if the grocer spends$250 advertising on the radio and \$500 advertising in the newspaper for a given month.
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$y=a{x}_{1}+b{x}_{2}+c$
We have to determine the function:
$\left\{\begin{array}{l}2400a+800b+c=36000\\ 2000a+500b+c=30000\\ 3000a+1000b+c=44000\end{array}$
Set up a system of equations for a,b,c:
$\left[\begin{array}{ccccc}2400& 800& 1& |& 36000\\ 2000& 500& 1& |& 30000\\ 3000& 1000& 1& |& 44000\end{array}\right]$
b) Build the augmented matrix:
$\left[\begin{array}{ccccc}1& 0& 0& |& 12\\ 0& 1& 0& |& 4\\ 0& 0& 1& |& 4000\end{array}\right]$
Use a graphing utility to find the reduced row-echelon form of the augmented matrix: $a=12,b=4,c=4000$
c) Determine a,b,c:
$y=12{x}_{1}+4{x}_{2}+4000$
write the model y that fits the data:
$y=12\left(2500\right)+4\left(500\right)+4000=36000$
d) Determine y for ${x}_{1}=2500,{x}_{2}=500$