Question

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

Answer 1

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 income

The 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 expenses

The 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 table

Using 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 expenses

See attachment

The pattern by which theatre's income and expenses increase

The 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 profit

This 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 inequalities

We 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

Read more about linear equations and inequalities at:

brainly.com/question/11234618