Question

The Enormous State University History Department offers three courses—Ancient, Medieval, and Modern History—and the chairperson is trying to decide how many sections of each to offer this semester. The department is allowed to offer 46 sections total, there are 5,300 students who would like to take a course, and there are 62 professors to teach them. Sections of Ancient History have 100 students each, sections of Medieval History hold 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. How many sections of each course should the chair schedule in order to offer all the sections that they are allowed, accommodate all of the students, and give one teaching assignment to each professor? HINT [See Example 2.] (Assume each section has the maximum number of students allowed.) Ancient History Medieval History Modern History

          The Enormous State University History Department offers three
courses—Ancient, Medieval, and Modern History—and the chairperson
is trying to decide how many sections of each to offer this
semester. The department is allowed to
offer 46 sections total, there
are 5,300 students who would like to take a course, and
there are 62 professors to teach them. Sections of
Ancient History have 100 students each, sections of Medieval
History hold 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. How many sections of each course should the
chair schedule in order to offer all the sections that they are
allowed, accommodate all of the students, and give one teaching
assignment to each professor? HINT [See Example 2.] (Assume each
section has the maximum number of students allowed.)
Ancient History  
Medieval History
Modern History

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Added by Elizabeth E.

Precalculus with Limits

Ron Larson 2nd Edition

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Solved by Expert Brent Burkett

12/22/2021

Step 1

**Step 1: Set up the variables and equations** Let X = number of sections of Ancient History Let Y = number of sections of Medieval History Let Z = number of sections of Modern History Show more…

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The Enormous State University History Department offers three courses—Ancient, Medieval, and Modern History—and the chairperson is trying to decide how many sections of each to offer this semester. The department is allowed to offer 46 sections total, there are 5,300 students who would like to take a course, and there are 62 professors to teach them. Sections of Ancient History have 100 students each, sections of Medieval History hold 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. How many sections of each course should the chair schedule in order to offer all the sections that they are allowed, accommodate all of the students, and give one teaching assignment to each professor? HINT [See Example 2.] (Assume each section has the maximum number of students allowed.) Ancient History Medieval History Modern History

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00:01 All right, so in this problem, there's a school trying to figure out how to schedule or how many sections to offer.

00:06 So you can read the question.

00:09 It's there.

00:10 So here's what i did.

00:12 So i set up my variables, x equals the number of ancient history courses, sections.

00:22 Y is equal to the number of medieval sections.

00:28 And z is equal to the number of modern history sections.

00:34 And then i started making up some equations.

00:39 So i know there's only 46 total sections.

00:43 So x plus y plus z has to equal or be less than or equal to 46.

00:52 Okay.

00:54 Based on the number of professors, since ancient history only needs, needs one professor, i said x, and medieval only is one professor, so i said y.

01:05 But there's two professors needed for modern history, so two for every one of those modern sections.

01:15 And that's got to equal 62.

01:18 And now we base everything off the number of students.

01:22 So ancient history can hold 100, so that's 100x plus, and then the medieval can hold 50, so 50 y, and then the modern history can hold 200, 200 z and so that's the total of all the students, which is 5 ,300, or 5 ,300 students.

01:51 All right, so first equation is the sections.

01:56 This one came from the number of professors, that are available.

02:01 And this last one is students.

02:04 How many students? all right.

02:07 So we have three equations and three unknowns.

02:09 Now we just got to take them and try and chunk them down until we get the eliminate variables, find x, y, z.

02:17 That's what we need.

02:17 Okay.

02:19 So i'm going to start with these first two.

02:23 I'm going to take, for the professors, x plus y plus 2z, equal 60, and for the sections, x plus y plus z equals 46, and i'm going to subtract those because the x's will disappear, the y's will disappear, and 2z minus z is just z.

02:46 62 minus 46 is 16.

02:51 So that means z is going to be 16, so we should offer 16 sections of modern history.

03:01 Well, now that we know that z is 16, i could plug it in everywhere if i want...

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