question_answer

If \[3{{x}_{1}}+5{{x}_{2}}\le 15\], \[5{{x}_{1}}+2{{x}_{2}}\le 10\], \[{{x}_{1}},\ {{x}_{2}}\ \ \ge 0\]                 then the maximum value of \[5{{x}_{1}}+3{{x}_{2}}\], by graphical method is

A)                 \[12\frac{7}{19}\]            

B)                 \[12\frac{1}{7}\]

C)                 \[12\frac{3}{5}\]              

D)                 12

Correct Answer: A

Solution :

                The shaded region represent the feasible region, hence \[p=5{{x}_{1}}+3{{x}_{2}}\]                 Obviously it is maximum at \[\left( \frac{20}{19},\frac{45}{19} \right)\]                                 Max \[p=5{{x}_{1}}+3{{x}_{2}}\]\[=5\left( \frac{20}{19} \right)+3\left( \frac{45}{19} \right)\]                                       \[=\frac{100}{19}+\frac{135}{19}=\frac{235}{19}=12\frac{7}{19}\].