10 points
Mary Ann pays a monthly fee for her cable package which includes a TV
movie a month. She gets billed an additional charge for every additional
movie. During her first month, Mary Ann watched 5 additional movies and
her bill was $135.50. During the second month, Mary Ann watched 9
additional movies and her bill was $157.50. Write a function that
represents the total monthly cost C(x) of Mary Ann's cable package,
where x is the number of additional movies watched. *

To model a real-life situation, a mathematical function can be constructed in such a way it accurately represents the variables measured in the system.

We need to build a function that represents the total monthly cost C(x) of Mary Ann's cable package. It consists of a fixed amount that includes one movie a month and a variable amount depending on the additional movies she watches. Let's call x to the additional movies, and represent the cost as a linear function of the form:

[tex]C(x)=C_o+bx[/tex]

Where Co and b are to be determined with the data provided.

In the first month, Mary Ann watched 5 additional movies and paid $135.50. This gives us an ordered pair (5,135.50). Substituting into the general function:

[tex]135.50=C_o+5b\qquad \qquad[1][/tex]

In the second month, Mary Ann watched 9 additional movies and paid $157.50. The point (9,157.50) is also used in the function:

[tex]157.50=C_o+9b\qquad \qquad[2][/tex]

Subtracting [2] and [1]:

[tex]157.50-135.50=4b=22[/tex]

Solving for b:

[tex]b=5.5[/tex]

Using this value in [1]:

[tex]135.50=C_o+5*5.5=C_o+27.50[/tex]

Solving for Co:

[tex]C_o=135.50-27.50=108[/tex]

The function that represents the total monthly cost C(x) of Mary Ann's cable package is:

[tex]\boxed{C(x)=108+5.5x}[/tex]