Problem 44 - Algebra 2

Question

For Exercises \(43-45,\) use the following information.The girls' softball team is sponsoring a fund-raising trip to see a professional baseball game. They charter a \(60-\) passenger bus for \(\$ 525 .\) In order to make a profit, they will charge \(\$ 15\) per person if all seats on the bus are sold, but for each empty seat, they will increase the price by \(\$ 1.50\) per person. What is the minimum number of passengers needed in order for the softball team not to lose money?

Step-by-Step Solution

Answer

The minimum number of passengers needed is 55.

Step 1: Define Variables

Let \( x \) be the number of empty seats on the bus. So, the number of passengers is \( 60 - x \). The price per ticket, then, will be \( 15 + 1.5x \). Our goal is to find the minimum number of passengers required to at least break even on the \$525 bus charter.

Step 2: Set Up the Revenue Equation

The total revenue can be calculated by multiplying the number of passengers by the ticket price per passenger. Therefore, the revenue \( R \) can be expressed as follows:\[ R = (60 - x)(15 + 1.5x) \]

Step 3: Ensure Revenue Covers Cost

For the team not to lose money, the revenue must be at least equal to the cost of the bus. Thus, we set up the inequality:\[ (60 - x)(15 + 1.5x) \geq 525 \]

Step 4: Simplify the Inequality

Expand the equation in the inequality:\[ (60 - x)(15 + 1.5x) = 60 \times 15 + 60 \times 1.5x - x \times 15 - 1.5x^2 \]\[ = 900 + 90x - 15x - 1.5x^2 \]\[ = 900 + 75x - 1.5x^2 \]Substitute back into the inequality:\[ 900 + 75x - 1.5x^2 \geq 525 \]

Step 5: Rearrange and Solve the Quadratic Inequality

Rearrange the inequality:\[ -1.5x^2 + 75x + 900 - 525 \geq 0 \]\[ -1.5x^2 + 75x - 375 \geq 0 \]We can divide the entire inequality by -1.5 (note this will reverse the inequality sign):\[ x^2 - 50x + 250 \leq 0 \]Solve the quadratic equation \( x^2 - 50x + 250 = 0 \) using the quadratic formula where \( a = 1 \), \( b = -50 \), and \( c = 250 \):\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]\[ x = \frac{50 \pm \sqrt{2500 - 1000}}{2} \]\[ x = \frac{50 \pm \sqrt{1500}}{2} \]

Step 6: Calculate and Conclude Passenger Requirement

Estimate \( \sqrt{1500} \) which is approximately \( \sqrt{1521} = 39 \), hence,\[ x = \frac{50 \pm 39}{2} \]This gives the solutions \( x = \frac{89}{2} = 44.5 \) and \( x = \frac{11}{2} = 5.5 \). Since \( x \) must be a whole number, \( x \) should be less than or equal to 5. Thus, to break even, the number of passengers cannot be less than \( 60 - 5 = 55 \).

Key Concepts

Revenue Equation

In order to determine how much money is being made, understanding the revenue equation is essential. The revenue equation is simply the product of the ticket price and the number of passengers. In this scenario, the softball team charges \(15 per ticket, but they also increase the ticket price by \)1.50 for each empty seat. Given that the bus holds 60 passengers, if there are \( x \) empty seats, then there will be \( 60 - x \) passengers. Therefore, the ticket price per passenger becomes: \( 15 + 1.5x \).
The revenue, \( R \), is then expressed by the formula:

• \( R = (60 - x)(15 + 1.5x) \).

Using the formula ensures that all variable factors – the number of empty seats and the changing ticket prices – are captured accurately.

Ticket Pricing

Ticket pricing plays a crucial role in determining the overall revenue. Initially, each passenger is charged a base price of \(15. However, to encourage full occupancy, the price per ticket increases by \)1.50 for each empty seat. This pricing strategy intends to mitigate losses by increasing revenue for fewer passengers.
To illustrate, assume there are 10 empty seats. The ticket price will then become \( 15 + 1.5 \times 10 = 30 \).
Thus, optimizing ticket pricing involves calculating the optimal number of tickets sold to maximize revenue or to at least cover the costs.

Break-Even Analysis

Break-even analysis helps in understanding the minimum conditions required to avoid losses. In this particular exercise, the softball team's goal is to break even on the \(525 cost of hiring the bus. This means the revenue generated should be at least \)525.
The condition for breaking even is represented by the inequality:

• \( (60 - x)(15 + 1.5x) \geq 525 \)

Solving this inequality provides the necessary number of passengers required – ensuring that revenue equals or exceeds costs. Break-even analysis, therefore, is a crucial financial tool that determines the minimum sales needed to cover expenses.

Quadratic Formula

The quadratic formula is pivotal when solving quadratic inequalities or equations. Here, this formula helps find out how many seats can remain empty without causing the team to incur a loss.

• For a quadratic equation \( ax^2 + bx + c = 0 \), the quadratic formula is: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

In this problem, the quadratic equation derived is \( x^2 - 50x + 250 = 0 \) with coefficients \( a = 1 \), \( b = -50 \), and \( c = 250 \).
Using the quadratic formula, you assess the possible values of \( x \), determining the extent of emptiness the bus can sustain while still being profitable. This method is essential in contexts where product prices or quantities dynamically affect financial outcomes.