Question
A catering company charges $200 plus $15 per guest. Write and graph the function
C(p). What is the cost for 30 guests? If the budget is $500, how many guests can attend? What is
the realistic domain?
C(p). What is the cost for 30 guests? If the budget is $500, how many guests can attend? What is
the realistic domain?
Answers
GPT-4o mini
To express the cost function \( C(p) \) for the catering company, we need to account for both the fixed cost and the variable cost per guest. The function can be defined as:
\[
C(p) = 200 + 15p
\]
where \( C(p) \) is the total cost and \( p \) is the number of guests.
### Graphing the Function
To graph this function, we can calculate a few values:
- For \( p = 0 \):
\[
C(0) = 200 + 15 \times 0 = 200
\]
- For \( p = 10 \):
\[
C(10) = 200 + 15 \times 10 = 200 + 150 = 350
\]
- For \( p = 20 \):
\[
C(20) = 200 + 15 \times 20 = 200 + 300 = 500
\]
- For \( p = 30 \):
\[
C(30) = 200 + 15 \times 30 = 200 + 450 = 650
\]
- For \( p = 40 \):
\[
C(40) = 200 + 15 \times 40 = 200 + 600 = 800
\]
You can plot these points on a Cartesian plane where the x-axis represents the number of guests \( p \) and the y-axis represents the total cost \( C(p) \). The graph will be a straight line with a slope of 15 and a y-intercept of 200.
### Cost for 30 Guests
To find the cost for 30 guests, we substitute \( p = 30 \) into the function:
\[
C(30) = 200 + 15 \times 30 = 200 + 450 = 650
\]
So, the cost for 30 guests is **$650**.
### Number of Guests with a Budget of $500
To determine how many guests can attend within a budget of $500, we set the cost function equal to 500 and solve for \( p \):
\[
200 + 15p = 500
\]
Subtracting 200 from both sides gives:
\[
15p = 300
\]
Dividing by 15 results in:
\[
p = 20
\]
So, with a budget of $500, **20 guests** can attend.
### Realistic Domain
Since you cannot have a negative number of guests, the lowest number of guests \( p \) is 0. Additionally, there is no theoretical upper limit on the number of guests as long as the budget allows for it, but at some point, the costs will become practically unmanageable. Therefore, a realistic domain for the number of guests is:
\[
[0, \infty)
\]
However, in practical scenarios any budget constraint would provide a maximum number of guests based on that budget. For example, with a budget of $500, the upper limit is 20 guests as calculated above.
\[
C(p) = 200 + 15p
\]
where \( C(p) \) is the total cost and \( p \) is the number of guests.
### Graphing the Function
To graph this function, we can calculate a few values:
- For \( p = 0 \):
\[
C(0) = 200 + 15 \times 0 = 200
\]
- For \( p = 10 \):
\[
C(10) = 200 + 15 \times 10 = 200 + 150 = 350
\]
- For \( p = 20 \):
\[
C(20) = 200 + 15 \times 20 = 200 + 300 = 500
\]
- For \( p = 30 \):
\[
C(30) = 200 + 15 \times 30 = 200 + 450 = 650
\]
- For \( p = 40 \):
\[
C(40) = 200 + 15 \times 40 = 200 + 600 = 800
\]
You can plot these points on a Cartesian plane where the x-axis represents the number of guests \( p \) and the y-axis represents the total cost \( C(p) \). The graph will be a straight line with a slope of 15 and a y-intercept of 200.
### Cost for 30 Guests
To find the cost for 30 guests, we substitute \( p = 30 \) into the function:
\[
C(30) = 200 + 15 \times 30 = 200 + 450 = 650
\]
So, the cost for 30 guests is **$650**.
### Number of Guests with a Budget of $500
To determine how many guests can attend within a budget of $500, we set the cost function equal to 500 and solve for \( p \):
\[
200 + 15p = 500
\]
Subtracting 200 from both sides gives:
\[
15p = 300
\]
Dividing by 15 results in:
\[
p = 20
\]
So, with a budget of $500, **20 guests** can attend.
### Realistic Domain
Since you cannot have a negative number of guests, the lowest number of guests \( p \) is 0. Additionally, there is no theoretical upper limit on the number of guests as long as the budget allows for it, but at some point, the costs will become practically unmanageable. Therefore, a realistic domain for the number of guests is:
\[
[0, \infty)
\]
However, in practical scenarios any budget constraint would provide a maximum number of guests based on that budget. For example, with a budget of $500, the upper limit is 20 guests as calculated above.