Problem 13 - Applied Calculus

Question

A movie theater has fixed costs of $$\$ 5000$$ per day and variable costs averaging $$\$ 2$$ per customer. The theater charges $$\$ 7$$ 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

The theater needs more than 1000 customers per day to make a profit.

Step 1: Define the Cost Function

The total cost for the theater per day is composed of fixed costs and variable costs per customer. The fixed cost is \( \\(5000 \) per day, while the variable cost is \( \\)2 \) per customer. Therefore, the cost function \( C(x) \) can be expressed as: \[ C(x) = 5000 + 2x \]where \( x \) is the number of customers.

Step 2: Define the Revenue Function

The revenue is generated from selling tickets, each for \( \$7 \). Thus, the revenue function \( R(x) \) is given by: \[ R(x) = 7x \]where \( x \) is the number of customers (or tickets sold).

Step 3: Find the Break-Even Point

The break-even point is where the cost equals the revenue, i.e., \( C(x) = R(x) \). Therefore, solve the equation:\[ 5000 + 2x = 7x \]Simplifying gives:\[ 5000 = 5x \]\[ x = \frac{5000}{5} = 1000 \]The theater needs 1000 customers per day to break even.

Step 4: Determine Condition for Profit

To make a profit, the revenue must exceed the costs, i.e., \( R(x) > C(x) \). Therefore, \[ 7x > 5000 + 2x \]Solving gives:\[ 5x > 5000 \]\[ x > \frac{5000}{5} = 1000 \]Thus, more than 1000 customers per day is needed for a profit.

Step 5: Graph Cost and Revenue Functions

Plot the cost function \( C(x) = 5000 + 2x \) and the revenue function \( R(x) = 7x \) on the same axes. The break-even point, where \( x = 1000 \), is the intersection of the two lines. At this point, both functions output the same value, indicating no profit or loss. To graph, plot cost starting at \( C(0) = 5000 \) but steady with a variable cost addition of 2, and revenue starting at \( R(0) = 0 \), increasing with each ticket sold at $7.

Key Concepts

Cost Functions

In the world of break-even analysis, understanding cost functions is key. A cost function represents the total cost a business incurs in producing a certain number of units or servicing a set amount of customers. It is generally comprised of two main components: fixed costs and variable costs.
Fixed costs are constant, regardless of the number of customers or units served. For our movie theater, this amount is \(5000 per day, representing expenses like rent or utilities that don’t change with audience size.
Variable costs, however, do fluctuate with the number of customers. In this scenario, each customer incurs a \)2 cost, perhaps covering expenses like staffing or concessions.
The cost function, therefore, can be formulated as:

• **Fixed Costs (constant):** \(5000
• **Variable Costs (per customer):** \)2
• **Cost Function:** \( C(x) = 5000 + 2x \)

where \( x \) represents the number of customers. This equation helps us understand our total expenses as customer numbers change.

Revenue Functions

Revenue functions showcase how much money a business brings in from sales. It's a straightforward calculation: multiplying the number of items sold or customers served by the price per unit or ticket.
For the movie theater, revenue stems from ticket sales at \(7 each. A revenue function is crucial to determine how sales affect overall income.
To express this in a formula, the revenue function \( R(x) \) can be written as:

• **Revenue per Ticket/Customer:** \)7
• **Revenue Function:** \( R(x) = 7x \)

where \( x \) is the number of customers or tickets sold. This function shows the linear relationship between the number of tickets sold and the revenue earned.

Profit Calculation

Profit calculation is the motive behind any business decision, and break-even analysis plays a vital role in achieving it. To find out if a business is profitable, we compare the revenue earned to the costs incurred.
In simple terms, **Profit** is calculated as:
\[ \text{Profit} = \text{Revenue} - \text{Cost} \]
To analyze when a business breaks even, like the movie theater, set cost equal to revenue:

• **Equation for Break-Even Point:** \( R(x) = C(x) \)

This translates to:
\[ 7x = 5000 + 2x \]
Solving for \( x \), the number of customers needed to break even is 1000. Simply put, at this customer level, the business neither makes a profit nor incurs a loss.

For profit, however, we need:
\[ R(x) > C(x) \]
Hence, more than 1000 customers are needed each day. Understanding these concepts of cost, revenue, and profit helps businesses strategize to not only meet but exceed the breakeven points for financial success.