Question

Minimize \( Z = 3x + 9y \) under the following constraints: \[ x + 3y \leq 60, \quad x + y \geq 10, \quad x \leq y, \quad x \geq 0, \quad y \geq 0. \]

Hint:

For optimization problems, identify the feasible region and evaluate the objective function at each vertex to determine the minimum or maximum.

Answer 1

Tominimize\(Z=3x+9y\),plottheconstraintsonagraphandfindthefeasibleregion: 1.\(x+3y\leq60\)isalinewithinterceptsat\(x=60,y=20\). 2.\(x+y\geq10\)isalinepassingthrough\((10,0)\)and\((0,10)\). 3.\(x\leqy\)representstheregionbelow\(x=y\). 4.\(x\geq0,y\geq0\)restrictstothefirstquadrant. Thefeasibleregionisbounded,and\(Z=3x+9y\)isevaluatedatcornerpoints.Solvefor\(Z\)atthesevertices: \begin{itemize} \item\((0,10)\):\(Z=3(0)+9(10)=90\), \item\((0,20)\):\(Z=3(0)+9(20)=180\), \item\((30,10)\):\(Z=3(30)+9(10)=90+30=120\). \end{itemize} Theminimumvalueis\(Z=90\)at\((0,10)\).