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Study Guides > Finite Math

Reading: Mathematical Modeling Exercise (incomplete content)

Many business contexts can be modeled with quadratic functions. This is because the expressions representing the price (price per item), the cost (cost per item), and the quantity (number of items sold) are typically linear. The product of any two of those linear expressions will produce a quadratic expression that can be used as a model for the business context. The variables used in business applications are not as traditionally accepted as variables that are used in physics applications, but there are some obvious reasons to use for cost, for price, and for quantity (all lowercase letters). For total production cost, we often use for the variable, for total revenue, and for total profit (all uppercase letters). You have seen these formulas in previous lessons, but we will review them here since we use them in the next two lessons. Business Application Vocabulary Unit Price (Price per Unit): The price per item a business sets to sell its product, sometimes represented as a linear expression. Quantity: The number of items sold, sometimes represented as a linear expression. Revenue: The total income based on sales (but without considering the cost of doing business). Unit Cost (Cost per Unit) or Production Cost: The cost of producing one item, sometimes represented as a linear expression. Profit: The amount of money a business makes on the sale of its product. Profit is determined by taking the total revenue (the quantity sold multiplied by the price per unit) and subtracting the total cost to produce the items (the quantity sold multiplied by the production cost per unit): . The following business formulas will be used in this lesson:
  • Total Production Costs = (cost per unit)(quantity of items sold)
  • Total Revenue = (price per unit)(quantity of items sold)
  • Profit = Total Revenue - Total Production Costs
Now answer the questions related to the following business problem: A theater decided to sell special event tickets at per ticket to benefit a local charity. The theater can seat up to people, and the manager of the theater expects to be able to sell all seats for the event. To maximize the revenue for this event, a research company volunteered to do a survey to find out whether the price of the ticket could be increased without losing revenue. The results showed that for each increase in ticket price, fewer tickets would be sold.
  1. Let represent the number of $1.00 price-per-ticket increases. Write an expression to represent the expected price for each ticket.
  2. Use the survey results to write an expression representing the possible number of tickets sold.
  3. Using as the number of -ticket price increases and the expression representing price per ticket, write the function, , to represent the total revenue in terms of the number of -ticket price increases.