QUESTION $2($Continuous$-$variable Optimization, Constrained$).$ Tool to use: Linear programming.

For construction of a civil engineering facility, a contractor has found natural reserves of sand and gravel

at Bloomingdale and Valley Springs where he may purchase such material. The unit cost including

delivery from Bloomingdale and Valley Springs is $$5$ and $$7,$ respectively. After the material is brought to

the site, it is mixed thoroughly and uniformly, and contract specification state that the mix should

contain a minimum of $30\%$ sand.

A total volume of $100,000m^3$ of mixed material is needed for the project. The Bloomingdale Pit

contains $25\%$ sand and the Valley Springs Pit contains $60\%$ sand.

As the new young construction engineer on the project, you are asked to determine how much

material should be taken from each pit in order to minimize the cost of material.

$($a$)$ Define the decision variables

$($b$)$ Write the constraints in mathematical form

$($c$)$ Write the objective function in mathematical form

$($d$)$ Find the optimum solution using the graphical method $($how much material should the

contractor take from each pit in order to minimize the overall cost of the material?$)$

$($e$)$ If you had not been hired, the contractor would have used $60,000m^3$ from Bloomingdale and

$40,000m^3$ from Valley Springs. How much did you save the company by giving them your expert

advice $($your answer to the previous question$)?$

$($f$)$ Solve the problem using MS Excel Solver and attach a printout of the program output to your

homework submission. Clearly mark the optimal solution in the output using a highlighter.