The Enormous State University History Department offers thre

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 43 sections total, there are 4,700 students who would like to take a course, and there are 56 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= sections

Medieval History= sections

Modern History= sections

Suppose in one year, total revenues from digital sales of pop/rock, tropical (salsa/merengue/cumbia/bachata), and urban (reggaeton) Latin music in a certain country amounted to $21 million. Pop/rock music brought in twice as much as the other two categories combined and $9 million more than tropical music. How much revenue was earned from digital sales in each of the three categories?
pop/rock music$ million

tropical music$ million

urban Latin music$ million

Solution

1)

let Ancient History has x sections,Medieval History has y sections,Modern History has z sections

equations are

x+y+z = 43

100x+ 50y + 200z = 4700

x+y+2z =56

x+y+2z- x-y-z =56-43

=>z= 13

x+y+z = 43 , z=13

=>x+y+13=43

=>x+y =30

=>y =30 - x

100x+ 50y + 200z = 4700, y =30-x , z=13

=>100x+ 50(30-x) + (200*13) = 4700

=>100x+ 1500-50x +2600 = 4700

=>50x = 4700-2600-1500

=>50x =600

=>x =12

y =30 - x , x =12

=>y =30 - 12

=>y =18

Ancient History= 12 sections

Medieval History=18 sections

Modern History=13 sections

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2)

let revenue earned from digital sales in pop/rock music$ x million,tropical music$ y million,urban Latin music$ z million

equations are

x+y+z=21

x=2y+2z

x=y+9

x=2y+2z => y+z =(x_/2)

x+y+z=21,y+z =(x_/2)

=>x+(x_/2)=21

=>(3x_/2) =21

=>x =21*2_/3

=> x =14

x=y+9, x=14

=>14=y+9

=>y=14-9

=>y=5

x+y+z=21,y =5,x=14

=>14+5+z=21

=>z=21-14-5

=>z =2

pop/rock music$ 14 million

tropical music$ 5 million

urban Latin music$ 2 million

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