Problem 19 - Applied Calculus

Question

A movie theater has fixed costs of $$ 5000$ per day and variable costs averaging $$ 6 per customer. The theater charges $$ 11 per ticket. (a) How many customers per day does the theater need in order to make a profit? (b) Find the cost and revenue functions and graph them on the same axes. Mark the break-even point.

Step-by-Step Solution

Answer

Theater needs more than 1000 customers for profit; break-even is at 1000 customers.

Step 1: Identify Fixed and Variable Costs

The fixed costs for the movie theater are \( \\(5000 \) per day. The variable cost is \( \\)6 \) per customer. These costs form the basis for our cost function.

Step 2: Formulate the Cost Function

The cost function, \( C(x) \), represents the total cost as a function of the number of customers, \(x\). It is calculated as the sum of fixed costs and variable costs: \[ C(x) = 5000 + 6x \] where \( x \) is the number of customers. This equation reflects how the total cost changes with more customers.

Step 3: Formulate the Revenue Function

The revenue function, \( R(x) \), represents the total revenue earned. Since the theater charges \( \$11 \) per ticket, the revenue is given by: \[ R(x) = 11x \] where \( x \) is the number of customers. This reflects how revenue increases as more customers attend.

Step 4: Find the Break-even Point

To find the break-even point, where costs equal revenue, set the cost function equal to the revenue function and solve for \(x\):\[ 5000 + 6x = 11x \]Simplify the equation:\[ 5000 = 5x \]\[ x = \frac{5000}{5} = 1000 \]This means the theater needs 1000 customers per day to break even.

Step 5: Determine Conditions for Profit

For the theater to make a profit, revenue must exceed costs; hence, \( R(x) > C(x) \). Solving:\[ 11x > 5000 + 6x \]\[ 5x > 5000 \]\[ x > 1000 \] Thus, more than 1000 customers per day are required for profit.

Step 6: Graph the Cost and Revenue Functions

Graph \( C(x) = 5000 + 6x \) and \( R(x) = 11x \) on the same axes. Mark the break-even point at (1000, 11000) since both functions equal \( \$11000 \) at this point. The area where the revenue curve is above the cost curve represents the profit region.

Key Concepts

Cost Function

The cost function is a mathematical expression that helps us understand how total costs change with the number of customers in a business. In the case of this movie theater, we have two types of costs: fixed costs and variable costs. Fixed costs remain constant regardless of the number of customers. Here, it's given as \( \\(5000 \) per day. Variable costs, on the other hand, change with the number of customers, and for this theater, it's \( \\)6 \) per customer.
To formulate the cost function \( C(x) \), we sum both fixed and variable costs. The function is:

• \( C(x) = 5000 + 6x \)

where \( x \) represents the number of customers. This function tells us that for each additional customer, the theater's total cost increases by \( \$6 \). Understanding the cost function is crucial for the theater to anticipate expenses and budget accordingly. The more customers attend, the higher the costs, but only up to the variable portion.

Revenue Function

The revenue function is a simple yet critical concept in break-even analysis. It reflects the total income a business generates from selling its products or services. For this theater example, revenue is earned through ticket sales. The theater charges \( \$11 \) per ticket. Therefore, if \( x \) represents the number of customers, the revenue function \( R(x) \) is:

• \( R(x) = 11x \)

This representation means that revenue increases directly with the number of customers. The more tickets sold, the more revenue the theater earns.
It's essential for businesses to track their revenue to gauge performance and achieve targets. By knowing how much a theater earns per customer, they can plan their resources better and strategize to increase attendance through promotions or other marketing efforts.

Profit Calculation

Profit calculation is the ultimate goal for many businesses, including our movie theater example. Profit occurs when revenue exceeds costs. To perform a profit calculation, we need to compare the revenue and cost functions.
First, we determine the break-even point, where costs equal revenue. Solving the equation \( C(x) = R(x) \), we find the number of customers needed to cover all costs. In this case:

• Set \( C(x) = R(x): \)
• \( 5000 + 6x = 11x \)
• \( 5000 = 5x \)
• \( x = 1000 \)

This means 1000 customers per day are required to break even.
For profit, we need more than break-even customers:

• \( 11x > 5000 + 6x \)
• \( 5x > 5000 \)
• \( x > 1000 \)

Thus, more than 1000 customers a day is required. Profit calculations are beneficial for strategizing and setting realistic goals to keep the business thriving.