The Enormous State University History Department offers thre

The Enormous State University History Department offers three courses, Ancient, Medieval, and Modern History, and the department chairperson is trying to decide how many sections of each to offer this semester. Please answer all parts of the question!!!

The Enormous State University History Department offers three courses, Ancient, Medieval, and Modern History, and the department chairperson is trying to decide how many sections of each to offer this semester. They may offer up to 54 sections total, up to 6,000 students would like to take a course, and there are 72 professors to teach them (no student will take more than one history course, and no professor will teach more than one section). Sections of Ancient History have 100 students each, sections of Medieval History have 50 students each, and sections of Modern History have 200 students each. Modern History sections are taught by a team of two professors, while Ancient and Medieval History need only one professor per section. Ancient History nets the university $2,000 per section, Medieval nets $4,000, and Modern History nets $6,000 per section. How many sections of each course should the department offer in order to generate the largest profit? section (s) Ancient History section (s) Medieval History section (s) Modern History What is the largest profit possible? Will there be any unused time slots, any students who did not get into classes, or any professors without anything to teach? Yes No

Solution

Let the number of the sections of Ancient History, Medieval History and Modern History be x, y and z respectively. Then we have x + y +z 54…(1), 100x+50y + 200z 6000 or, on dividing both the sides by 50, we get 2x+y+4z 120 …(2) and x+y +2z 72…(3). On solving these inequalities, we get x 12, y 24 and z 18.

The University’s revenue function is 2000x +4000y + 6000z. Assuming that the infrastructural and the other costs are similar for all the sectiions, it is apparent that the university’s revenue, and hence its profit, will be maximum when z is maximum, followed by y. This means that z = 18, y = 24 and x = 12.

Thus, the maximum number os sections are as under:

Ancient History       12

Medieval History    24

Modern History       18

The largest revenue, hence the largest profit possible, is 2000*12 +4000*24 + 6000*18 = 24000+96000+108000 = $228000.

There will not be any unused time slots and also there will not be any students who did not get into classes and nor will there be any professors who did not get anything to teach. The answer is NO.