Question

Widget wonders produces widgets. they have found that the cost, c(x), of making x widgets is a quadratic function in terms of x.....the company also discovered that it costs $103. to produce 3 widgets, $163 to produce 7 widgets, and $328 to produce 12 find the total cost of producing 2 widgets

Answer 1

$c(x)=ax^2+bx+c \\ \\ c(3)=103 \\ c(7)=163 \\ c(12)=328 \\ \\ a \times 3^2+b \times 3 + c=103 \\ a \times 7^2 + b \times 7 + c= 163 \\ a \times 12^2 + b \times 12 + c =328 \\ \\ 9a+3b+c=103 \\ 49a+7b+c=163 \\ 144a+12b+c=328$

$\hbox{the first two equations:} \\ 9a+3b+c=103 \ \ \ \ \ |\times (-1) \\ 49a+7b+c=163 \\ \\ -9a-3b-c=-103 \\ \underline{49a+7b+c=163 \ \ \ \ \ } \\ 40a+4b=60 \ \ \ \ \ \ \ \ \ \ \ |\div 4 \\ 10a+b=15$

$\hbox{the second and third equation:} \\ 49a+7b+c=163 \ \ \ \ \ \ \ |\times (-1) \\ 144a+12b+c=328 \\ \\ -49a-7b-c=-163 \\ \underline{144a+12b+c=328 \ \ } \\ 95a+5b=165 \ \ \ \ \ \ \ \ \ \ \ \ |\div 5 \\ 19a+b=33$

$\hbox{set up a new system of equations:} \\ 10a+b=15 \ \ \ \ \ \ \ \ |\times (-1) \\ 19a+b=33 \\ \\ -10a-b=-15 \\ \underline{19a+b=33 \ \ \ \ \ \ } \\ 9a=18 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ |\div 9 \\ a=2 \\ \\ 10a+b=15 \\ 10 \times 2+b=15 \\ 20+b=15 \ \ \ \ \ \ \ \ \ \ |-20 \\ b=-5$

$49a+7b+c=163 \\ 49 \times 2+7 \times (-5)+c=163 \\ 98-35+c=163 \\ 63+c=163 \ \ \ \ \ \ \ \ \ \ \ \ \ \ |-63 \\ c=100 \\ \\ \hbox{the equation for the cost of maxing x widgets:} \\ c(x)=2x^2-5x+100 \\ \\ c(2)=2 \times 2^2-5 \times 2+100=8-10+100=98$

The cost of producing 2 widgets is $98.