Find the total cost of producing 6 widgets.
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 $23 to produce 2 widgets, $103 to produce 4 widgets, and $631 to produce 10 widgets.
If c(x) is a quadratic function, then $${c}{\left({\mathtt{x}}\right)} = {\mathtt{A}}{\mathtt{\,\times\,}}{{\mathtt{x}}}^{{\mathtt{2}}}{\mathtt{\,\small\textbf+\,}}{\mathtt{B}}{\mathtt{\,\times\,}}{\mathtt{x}}{\mathtt{\,\small\textbf+\,}}{\mathtt{C}}$$, where A, B, and C are constants. Our job here is to find the constants.
$$c(2) = 23 = A*2^2 + B*2 + C$$
$$c(4) = 103 = A*4^2 + B*4 + C$$
$$c(10) = 631 = A*10^2 + B*10 + C$$
So we have:
$$23 = 4A + 2B + C$$
$$103 = 16A + 4B + C$$
$$631 = 100A + 10B + C$$
If we solve that, we get A = 6, B = 4, C = -9.
Thus, since $${c}{\left({\mathtt{x}}\right)} = {\mathtt{A}}{\mathtt{\,\times\,}}{{\mathtt{x}}}^{{\mathtt{2}}}{\mathtt{\,\small\textbf+\,}}{\mathtt{B}}{\mathtt{\,\times\,}}{\mathtt{x}}{\mathtt{\,\small\textbf+\,}}{\mathtt{C}}$$:
$$c(x) = 6x^2+4x-9$$
Now we just use the function for x=6.
$${c}{\left({\mathtt{6}}\right)} = {\mathtt{6}}{\mathtt{\,\times\,}}{{\mathtt{6}}}^{{\mathtt{2}}}{\mathtt{\,\small\textbf+\,}}{\mathtt{4}}{\mathtt{\,\times\,}}{\mathtt{6}}{\mathtt{\,-\,}}{\mathtt{9}}$$
$$c(6) = 231$$ dollars.
If c(x) is a quadratic function, then $${c}{\left({\mathtt{x}}\right)} = {\mathtt{A}}{\mathtt{\,\times\,}}{{\mathtt{x}}}^{{\mathtt{2}}}{\mathtt{\,\small\textbf+\,}}{\mathtt{B}}{\mathtt{\,\times\,}}{\mathtt{x}}{\mathtt{\,\small\textbf+\,}}{\mathtt{C}}$$, where A, B, and C are constants. Our job here is to find the constants.
$$c(2) = 23 = A*2^2 + B*2 + C$$
$$c(4) = 103 = A*4^2 + B*4 + C$$
$$c(10) = 631 = A*10^2 + B*10 + C$$
So we have:
$$23 = 4A + 2B + C$$
$$103 = 16A + 4B + C$$
$$631 = 100A + 10B + C$$
If we solve that, we get A = 6, B = 4, C = -9.
Thus, since $${c}{\left({\mathtt{x}}\right)} = {\mathtt{A}}{\mathtt{\,\times\,}}{{\mathtt{x}}}^{{\mathtt{2}}}{\mathtt{\,\small\textbf+\,}}{\mathtt{B}}{\mathtt{\,\times\,}}{\mathtt{x}}{\mathtt{\,\small\textbf+\,}}{\mathtt{C}}$$:
$$c(x) = 6x^2+4x-9$$
Now we just use the function for x=6.
$${c}{\left({\mathtt{6}}\right)} = {\mathtt{6}}{\mathtt{\,\times\,}}{{\mathtt{6}}}^{{\mathtt{2}}}{\mathtt{\,\small\textbf+\,}}{\mathtt{4}}{\mathtt{\,\times\,}}{\mathtt{6}}{\mathtt{\,-\,}}{\mathtt{9}}$$
$$c(6) = 231$$ dollars.
