1. Suppose that a theater charges a school group $4.50 per student to show a special film. Suppose that the theater's operating expenses include $130 for the staff and a film rental fee of $1.25 per student.
a. What equation relates the number of students x to the theater's income I?
b. What equation relates the theater's operating expenses E to the number of students x?
c. Copy and complete the table below.
Theater Income and Expenses
Number of Students, x 0 10 20 30 40 50 60 70
Income, I ($) box box box box box box box box
Expenses, E ($) box box box box box box box box
d. On the same set of axes, graph the theater's income and operating expenses for any number of students from 0 to 100.
e. Describe the patterns by which income and operating expenses increase as the number of students in a group increases.
f. Write and solve an equation that you can use to answer the question “How many students need to attend the movie so that the theater's income will equal its operating expenses?”
g. Write an equation that represents the theater's profit. Compare your equation to those your classmates wrote.
h. Find the number of students that make each of the following inequality statements true.
i. E < 255
ii. I > 675
Answers
The functions I(x) and E(x) of the theatre's income and expenses are illustrations of linear functions.
The equation of the theatre's incomeThe theatre charges $4.50 per student.
Assume the number of students is x, the equation of the theatre's income would be:
I(x) = 4.5x
The equation of the theatre's expensesThe theatre expense per student is $1.25, and the operating cost on the staff is $130
The equation of the theatre's expenses would be:
E(x) = 1.25x + 130
Complete the tableUsing the formulas I(x) = 4.5x and E(x) = 1.25x + 130, the complete table is:
Students, x 0 10 20 30 40 50 60 70
Income, I 0 45 90 135 180 225 270 315
Expenses, E 130 142.5 155 167.5 180 192.5 205 217.5
The graph of the theatre's income and expensesSee attachment
The pattern by which theatre's income and expenses increaseThe functions I(x) and E(x) are linear functions.
So, the pattern with which the functions increase is a linear pattern.
The number of students when the theatre's income and expenses are equal
This means that:
I(x) = E(x)
So, we have:
4.5x = 1.25x + 130
Subtract 1.25 from both sides
3.25x = 130
Divide both sides by 3.25
x = 40
Hence, the number of students is 40
The theatre profitThis is the difference between the theatre expenses and their income.
So, we have:
P(x) = E(x) - I(x)
This gives
P(x) = 1.25x + 130 - 4.5x
Simplify
P(x) = 130 - 3.25x
Solution to the inequalitiesWe have:
E(x) < 255
This gives
1.25x + 130 < 255
Subtract 130 from both sides and divide by 1.25
x < 100 students
Also, we have:
I(x) > 675
This gives
4.5x > 657
Solve for x
x > 146 students
Hence, the number of students for the inequalities are less than 100 and greater than 146
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Remark
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2/61/3[tex]\qquad\qquad\huge\underline{{\sf Answer}}♨[/tex]
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Which choice shows the graph of the solution set for the inequality p - 2 > 6?
(A
"10
"5
0
5
10
B)
10
5
0
5
10
"10
5
HHHO+++
5 10
0
*10
-5
HHHO++++
0 5 10
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Add 2 to each side
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