Case Study $2(10\mathrm{.}5$ Points$)$

The Queen City Nursery manufactures bags of potting soil from compost and topsoil. Each cubic foot of compost costs $20$ cents and contains $4$ pounds of sand, $3$ pounds of clay, and $5$ pounds of humus. Each cubic foot of topsoil costs $30$ cents and contains $3$ pounds of sand, $5$ pounds of clay, and $7$ pounds of humus. Each bag of potting soil must contain at least $12$ pounds of sand, at least $15$ pounds of clay, and at least $10$ pounds of humus.

A$.$ How many cubic feet of compost and topsoil should be bought by Queen City Nursery so that the cost per bag is kept to a minimum level? Use the graphical LP approach. $($Show your work both manually and using Excel Solver$)$

B$.$ At the optimal solution, calculate the surplus for each constraint?