MathProblem 13 Wine and cigar problem

Joe's happiness is proportional to the function \(w^2*c\) (\(w\) squared * \(c\)), where \(w\) stands for daily consumption of glasses of wine, and \(c\) stands of daily consumption of number of cigars. Wine costs \(3\) a glass and cigars cost \(2\) each. His daily budget for both is $100. How should he divide his money between wine and cigars to maximize his happiness? Assume he does not have to buy integer numbers of either cigars or wine.

Solution

很简单的优化问题。我们需要 \(\max f= c\cdot w^2\)，唯一的限制是

\[3w+2c=100 \]

变换一下得到：

\[c\cdot w^2 = w^2(100-3w)/2 \]

求导即可解决。