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Catering a dinner: You are having a dinner catered. You pay a rental fee of \(\$ 150\) for the dining hall, and you pay the caterer \(\$ 10\) for each person who attends the dinner. a. Suppose you just want to break even. i. How much should you charge per ticket if you expect 50 people to attend? ii. Use a formula to express the amount you should charge per ticket as a function of the number of people attending. Be sure to explain the meaning of the letters you choose and the units. iii. You expect 65 people to attend the dinner. Use your answer to part ii to express in functional notation the amount you should charge per ticket, and then calculate that amount. b. Suppose now that you want to make a profit of \(\$ 100\) from the dinner. Use a formula to express the amount you should charge per ticket as a function of the number of people attending. Again, be sure to explain the meaning of the letters you choose and the units.

Step by step solution

01

Understanding the Total Cost at Break-Even

At break-even, you need to cover the rental fee and catering cost. The rental fee is \( \\(150 \), and you pay \( \\)10 \) per person attending. If \( n \) is the number of attendees, then the total cost is \( 150 + 10n \).

02

Determining Charge Per Ticket for 50 People

To break even with 50 people attending, the total revenue must equal the total cost. The equation is \( 50x = 150 + 10 \times 50 \), where \( x \) is the ticket price. Solving for \( x \) gives \( x = 13 \).

03

Formulating General Cost Per Ticket

Express the ticket price as a function of attendees, \( f(n) = \frac{150 + 10n}{n} \), where \( f(n) \) is the charge per ticket and \( n \) is the number of people.

04

Calculating Specific Ticket Price for 65 People

Using \( f(n) = \frac{150 + 10n}{n} \), substitute \( n = 65 \) to get \( f(65) = \frac{150 + 10 \times 65}{65} = 12.31 \).

05

Formula for Targeting a Profit

To make a \( \$100 \) profit, the total revenue needs to be \( 250 + 10n \). Use \( g(n) = \frac{250 + 10n}{n} \) where \( g(n) \) is the ticket price ensuring a profit.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Break-even analysis

The concept of break-even analysis is crucial for understanding how to cover costs in any financial venture. To break even means that the total costs equal the total revenues, leaving no profit or loss.
In the context of our dinner example, we need to account for fixed costs such as the rental fee of $150, and variable costs based on attendance, which is $10 per person. By calculating total costs for a given number of attendees, we can determine the price per ticket required to achieve break-even. This involves setting the total revenue (number of tickets sold times ticket price) equal to the total cost.
Understanding break-even helps businesses and individuals make informed pricing decisions and assess their operational efficiency.

Cost function

A cost function is a mathematical formula that helps us calculate the total cost based on different variables, like the number of units produced or, in this case, the number of attendees.
In our scenario, the total catering cost can be expressed as: \[ C(n) = 150 + 10n \] Here, \( C(n) \) is the total cost when \( n \) is the number of people attending the dinner.
The fixed cost is undeniably the rental fee of \(150. The variable part depends on the number of attendees, represented by \)10 multiplied by \( n \), which is their individual cost contribution.

Profit function

To calculate for profit, not only do costs need to be covered, but additional revenue should be generated. For this, a profit function is used to set an income target beyond breaking even.
In the example, suppose we want to make a $100 profit. The revenue must now equal the total cost plus this desired profit. Thus, the formula evolves to: \[ R(n) = 250 + 10n \] Here, \( R(n) \) stands for the revenue needed to achieve the desired profit for \( n \) attendees.
By solving for the ticket price in this adjusted equation, we ensure calculated profits are attainable when reaching a target number of guests.

Ticket pricing

Determining the right ticket price is essential for balancing cost recovery and achieving profit goals.
To break even with \( n \) attendees, we derive a price per ticket using: \[ P(n) = \frac{150 + 10n}{n} \] This equation helps find the ticket price that ensures all costs are met without gaining or losing money.
If a profit is desired, the ticket pricing function adjusts to: \[ P'(n) = \frac{250 + 10n}{n} \] Utilizing these functional equations, one can explore different price strategies by considering potential attendance, thereby predicting financial outcomes and ensuring business success.