Resource allocation. A coffee manufacturer uses Colombian and Brazilian coffee beans to produce two​ blends, robust and mild. A pound of the robust blend requires 1212 ounces of Colombian beans and 44 ounces of Brazilian beans. A pound of the mild blend requires 66 ounces of Colombian beans and 1010 ounces of Brazilian beans. Coffee is shipped in 8080​-pound burlap bags. The company has 5151 bags of Colombian beans and 3333 bags of Brazilian beans on hand. How many pounds of each blend should they produce in order to use all the available​ beans?

Accepted Solution

The company inventory consists of 51 bags of Colombian and 33 bags of Brazilian beans. Each bag holds 80 pounds of beans, so in total the company has 4080 pounds of Colombian and 2640 pounds of Brazilian beans.The company wants to use up its entire inventory, a total of 6720 pounds of beans.Let [tex]r[/tex] and [tex]m[/tex] denote the amount (in pounds) of the robust and mild blends, respectively, that the company should end up producing.To use the entire inventory, we must have[tex]r+m=6720[/tex]Each pound of the robust blend uses 12 ounces (3/4 = 0.75 pound) of Colombian beans, and each pound of the mild blend uses 6 ounces (3/8 = 0.375 pound) of Colombian beans, so that[tex]0.75r+0.375m=4080[/tex]while each pound of the robust blend uses 4 ounces (1/4 = 0.25 pound) of Brazilian beans, and each pound of the mild blend uses 10 ounces (5/8 = 0.625 pound) of Brazilian beans, so that[tex]0.25r+0.625m=2640[/tex]Multiply both equations by 8 to get rid of the rational coefficients:[tex]\begin{cases}6r+3m=32640\\2r+5m=21120\end{cases}[/tex]Subtract 3(second equation) from (first equation) to eliminate [tex]r[/tex]:[tex](6r+3m)-3(2r+5m)=32640-3\cdot21120[/tex][tex]-12m=-30720\implies\boxed{m=2560}[/tex]Then[tex]r+2560=6720\implies\boxed{r=4160}[/tex]So the company needs to produce 4160 pounds of the robust blend and 2560 pounds of the mild blend.