A new engineering building at a state university is to contain \(10,000,000\) square feet. The total cost of the building (TC) is given by the following expression:

\[\mathrm{TC}=\left(200+80 X+2 X^{2}\right) A\]

where \(X=\) number of floors and \(A=\) floor area in \(\mathrm{ft}^{2} /\) floor.

a. Create a table that shows the total building cost, average cost per floor, and marginal cost per floor (using the difference equation approach) for configurations ranging from 1 floor to 12 floors, inclusive. (Hint: You may want to create the table using your favorite spreadsheet program.)

b. Based on your table, what is the optimal number of floors for the building? Justify your answer based on the "total cost'" column.

c. Justify your answer in Part b based on the "marginal cost per floor" column.

d. Demonstrate, using differential calculus, that your answer in Parts b and c is correct. (Note: For this part, assume that \(X\) is a continuous variable.)