Question

Mia is going out to eat with some friends and has a total of $15 to spend. She got a water to drink and plans to leave a $3 tip. The price of the entrees range from $5.98 to $24.99. She must also remember that there is a 7% sales tax applied to the cost of her meal. The total cost she pays for her meal, including the tip, can be modeled by the function f(c), where c represents the price of the entree she orders.

The domain of the function is__ ≤ c ≤ __.

Mia's friend Clara loans her $5 so that Mia has more entrees to choose from on the menu. With this additional $5, the domain of the function representing the price of an entree Mia can afford is now __≤ c ≤ __.

Answer 1

1. First we are going to find the cost function.
We know for our problem that there is a 7% sales tax applied to the cost of her meal and she plans to leave 3$ as tip. Let
`c` represent the cost of the meal:

`f(c)=c+0.07c+3`

`f(c)=1.07c+3`
where

`c` is the cost of the entree

`f(c)` is the total cost she will pay

We know that the minimum cost of an entree is $5.98, so the domain of our functions so far is:
`5.98 \leq c \leq`
Now, to find the upper limit of the domain, we are going to take advantage of the fact the she only has $15 to spend, so we can replace the total cost with 15 and solve for
`c`:

`f(c)=1.07c+3`

`15=1.07c+3`

`1.07c=12`

`c= (12)/(1.07)`

`c=11.21`

We can conclude that the domain of the function is
`5.98 \leq c \leq 11.21`

2. We know that Mia's friend Clara loans her $5, so Mia's total money now is $15+$5=$20. Since the minimum cost for an entree remains the same ($5.98), the lower limit of our domain remains the same:
`5.98 \leq c \leq`
Now, to find the upper limit of our domain, we are going to replace the total cost with 20 an solve for
`c`:

`f(c)=1.07c+3`

`20=1.07c+3`

`1.07c=17`

`c= (17)/(1.07)`

`c=15.88`

We can conclude that the domain of the function representing the price of an entree Mia can afford is now
`5.98 \leq c \leq 15.88`