Question
Dr. Middleton is trying to create a formula with his student,
Jacobi, to solidify the fundraising goal for the Men of
Excellence organization’s gala based on the number of gala
attendees. The organization has raised $500 from pre-ticket
sales. The tickets are selling for $10 a ticket. The total cost
for the organization’s gala is represented by the equation C =
500 + 10n.
A. Interpret the equation that was given to you above. What does
the C represent in the equation? What does n represent in the
equation? Explain your reasoning.
B. Jacobi wants to ensure that the total cost for the gala does not
exceed $800. Model and solve a linear inequality that represents the
situation. What is the maximum number of attendees (n) that Jacobi
can have while staying within budget?
C. Dr. Middleton and Jacobi decided that for the organization to be
successful, they must have 30 attendees at the gala. Model and
write an inequality that represents this requirement. What is the
minimum amount that Jacobi and Dr. Middleton can spend to help
their fundraising goal if they have 30 members purchasing a ticket to
the gala?
Jacobi, to solidify the fundraising goal for the Men of
Excellence organization’s gala based on the number of gala
attendees. The organization has raised $500 from pre-ticket
sales. The tickets are selling for $10 a ticket. The total cost
for the organization’s gala is represented by the equation C =
500 + 10n.
A. Interpret the equation that was given to you above. What does
the C represent in the equation? What does n represent in the
equation? Explain your reasoning.
B. Jacobi wants to ensure that the total cost for the gala does not
exceed $800. Model and solve a linear inequality that represents the
situation. What is the maximum number of attendees (n) that Jacobi
can have while staying within budget?
C. Dr. Middleton and Jacobi decided that for the organization to be
successful, they must have 30 attendees at the gala. Model and
write an inequality that represents this requirement. What is the
minimum amount that Jacobi and Dr. Middleton can spend to help
their fundraising goal if they have 30 members purchasing a ticket to
the gala?
Answers
GPT-4o mini
Let's break down each part of your inquiry:
### A. Interpretation of the Equation
The equation given is \( C = 500 + 10n \).
- **C** represents the total cost of the gala. This encompasses all expenses that the organization has to pay in order to hold the gala.
- **n** represents the number of gala attendees (or the number of tickets sold).
**Reasoning:** The equation specifies that the total cost (C) is derived from a fixed base cost of $500 (which could include venue rental, catering, etc.) plus $10 for each ticket sold (which is the variable cost corresponding to each attendee). Therefore, the more attendees there are, the higher the total cost will be based on how many tickets are sold at $10 each.
### B. Modeling and Solving a Linear Inequality
Jacobi wants to make sure that the total cost for the gala does not exceed $800. We can set up the inequality as:
\[ C \leq 800 \]
Substituting the cost equation into the inequality, we have:
\[ 500 + 10n \leq 800 \]
Now, let's solve the inequality:
1. Subtract 500 from both sides:
\[ 10n \leq 800 - 500 \]
\[ 10n \leq 300 \]
2. Divide both sides by 10:
\[ n \leq 30 \]
**Maximum Number of Attendees (n):** Jacobi can have a maximum of 30 attendees while staying within budget.
### C. Inequality for Minimum Attendees and Cost
Dr. Middleton and Jacobi have decided they need at least 30 attendees. We can model this requirement with the inequality:
\[ n \geq 30 \]
To find the minimum amount that Jacobi and Dr. Middleton can expect to spend with 30 attendees, we substitute \( n = 30 \) into the cost equation \( C \):
\[ C = 500 + 10n \]
Substituting \( n = 30 \):
\[ C = 500 + 10 \times 30 \]
\[ C = 500 + 300 \]
\[ C = 800 \]
**Minimum Amount to Spend:** The minimum amount that Jacobi and Dr. Middleton can spend with 30 attendees purchasing tickets is $800.
### A. Interpretation of the Equation
The equation given is \( C = 500 + 10n \).
- **C** represents the total cost of the gala. This encompasses all expenses that the organization has to pay in order to hold the gala.
- **n** represents the number of gala attendees (or the number of tickets sold).
**Reasoning:** The equation specifies that the total cost (C) is derived from a fixed base cost of $500 (which could include venue rental, catering, etc.) plus $10 for each ticket sold (which is the variable cost corresponding to each attendee). Therefore, the more attendees there are, the higher the total cost will be based on how many tickets are sold at $10 each.
### B. Modeling and Solving a Linear Inequality
Jacobi wants to make sure that the total cost for the gala does not exceed $800. We can set up the inequality as:
\[ C \leq 800 \]
Substituting the cost equation into the inequality, we have:
\[ 500 + 10n \leq 800 \]
Now, let's solve the inequality:
1. Subtract 500 from both sides:
\[ 10n \leq 800 - 500 \]
\[ 10n \leq 300 \]
2. Divide both sides by 10:
\[ n \leq 30 \]
**Maximum Number of Attendees (n):** Jacobi can have a maximum of 30 attendees while staying within budget.
### C. Inequality for Minimum Attendees and Cost
Dr. Middleton and Jacobi have decided they need at least 30 attendees. We can model this requirement with the inequality:
\[ n \geq 30 \]
To find the minimum amount that Jacobi and Dr. Middleton can expect to spend with 30 attendees, we substitute \( n = 30 \) into the cost equation \( C \):
\[ C = 500 + 10n \]
Substituting \( n = 30 \):
\[ C = 500 + 10 \times 30 \]
\[ C = 500 + 300 \]
\[ C = 800 \]
**Minimum Amount to Spend:** The minimum amount that Jacobi and Dr. Middleton can spend with 30 attendees purchasing tickets is $800.