A construction project requires $8,000$ m$3$ of aggregate. The aggregate must contain no less than $40\%$ sand and no more than $55\%$ gravel. Materials may be obtained from two sites. Material from each site has a different mixture of sand and gravel, and comes at a different delivery cost. These are described below.

Delivery Cost

Site $($$$/$m$3)\%$ Sand $\%$ Gravel $153565$

$276040$

The objective is to find the least$-$cost mixture of material from the two sites.

a$)$ Formulate the problem as a linear program, with a linear objective function and linear constraints. Identify the decision variables. Explain each constraint.

b$)$ Graph the feasible region and solve the problem graphically.

c$)$ Solve the problem using Excel.

d$)$ Briefly interpret the results, and explain the meaning of each Lagrange multiplier.