Problem 59 - College Algebra with Corequisite Support

Question

For the following exercises, solve for the desired quantity. A musician charges \(C(x)=64 x+20,000\) where \(x\) is the total number of attendees at the concert. The venue charges $$\$ 80$$ per ticket. After how many people buy tickets does the venue break even, and what is the value of the total tickets sold at that point?

Step-by-Step Solution

Answer

The venue breaks even when 1,250 tickets are sold, generating \$100,000 in revenue.

Step 1: Define Variables and Equations

Let \( x \) represent the number of attendees. The cost \( C(x) \) for the musician to perform is given by the equation \( C(x) = 64x + 20,000 \). The revenue from ticket sales is calculated as \( R(x) = 80x \). To find the break-even point, set the revenue equation equal to the cost equation to solve for \( x \).

Step 2: Set Up the Break-Even Equation

Set the cost equation equal to the revenue equation: \[ 80x = 64x + 20,000 \].This equation represents the point where the income from ticket sales exactly covers the musician's fees.

Step 3: Solve for x

To find \( x \), the number of attendees needed to break even, solve the equation from Step 2: \[ 80x = 64x + 20,000 \]. Subtract \( 64x \) from both sides to isolate terms with \( x \): \[ 16x = 20,000 \]. Then, divide both sides by 16: \[ x = \frac{20,000}{16} = 1,250 \].

Step 4: Calculate Total Revenue

At \( x = 1,250 \), calculate the total revenue to verify the break-even point: \( R(x) = 80 \times 1,250 \). Upon calculation, we get \( R(1,250) = 100,000 \). Thus, the total revenue is \$100,000, confirming that the venue breaks even with 1,250 attendees.

Key Concepts

Cost Equation

When analyzing a break-even problem, the cost equation helps determine total expenses. In this exercise, the cost equation is given as \( C(x) = 64x + 20,000 \). This equation represents:

• The fixed cost: \( 20,000 \), which is constant regardless of the number of attendees. This amount could include expenses such as venue hire, equipment, and other overhead costs.
• The variable cost: \( 64x \), which depends on the number of attendees \( x \). In this case, it can include additional expenses like production or staffing that increase with the number of concert-goers.

By understanding both types of costs, you can see how expenses change as the number of attendees changes. The cost equation is crucial for determining the break-even point.

Revenue Equation

The revenue equation calculates the total income generated from sales. In this context, the equation is \( R(x) = 80x \). Breaking down what this represents:

• Ticket price: \( \$80 \) per ticket is the price set by the venue. This is the income received for each attendee.
• Total revenue: \( 80x \) signifies the overall earnings when \( x \) tickets are sold.

Revenue equations help predict income based on different attendance scenarios. It is essential to see how revenue scales with the number of tickets sold. This understanding is vital for further analysis, like finding the break-even point.

Solve for x

To find out when the venue breaks even, you need to solve the equation where income equals expense. This is done by equating the cost and revenue equations: \[ 80x = 64x + 20,000 \] Here's a simple breakdown of solving for \( x \):

• Subtract \( 64x \) from each side to get terms with \( x \) on one side: \( 16x = 20,000 \).
• Divide both sides by 16 to isolate \( x \): \( x = \frac{20,000}{16} = 1,250 \).

Thus, \( x = 1,250 \) means that 1,250 attendees are needed for revenues to match costs. At this point, no profit or loss is incurred, achieving the break-even point.

Ticket Pricing

Ticket pricing is a critical component that affects both revenue and the break-even point. In this scenario:

• Each ticket is priced at \( \$80 \).
• The higher the ticket price, the fewer attendees required to break even.
• Conversely, a lower ticket price demands more attendees to reach the same financial equilibrium.

The right ticket price balances attracting attendees and covering costs. In this exercise, the venue breaks even exactly when the total revenue matches the total costs at the set ticket price. Understanding this concept ensures financial decisions are well-informed and aligned with business goals.