Problem 87 - Algebra and Trigonometry

Question

Theater Tour Cost A riverboat theater offers bus tours to groups on the following basis. Hiring the bus costs the group \(\$ 360\) , to be shared equally by the group members. Theater tickets, normally \(\$ 30\) each, are discounted by 25\(€\) times the number of people in the group. How many members must be in the group so that the cost of the theater tour (bus fare plus theater ticket) is less than \(\$ 39\) per person?

Step-by-Step Solution

Answer

The group must have more than 24 members.

Step 1: Define the Cost of Bus per Person

The total cost for hiring the bus is \$360, which is shared equally among the group members. If we denote the number of group members by \(n\), the cost per person for the bus part is \(\frac{360}{n}\).

Step 2: Calculate the Discounted Ticket Price

The theater ticket price per person is originally \$30. The discount received is \(0.25 \times n\), where \(n\) is the number of people in the group. Thus, the discounted price of a theater ticket per person is \(30 - 0.25n\).

Step 3: Create an Equation for Total Cost per Person

Sum the bus cost per person and the discounted theater ticket cost per person to get the total cost per member: \[ \text{Total Cost per Person} = \frac{360}{n} + (30 - 0.25n) \]

Step 4: Set Up Inequality for Cost to be Less than $39

To find out when the cost is less than \$39 per person, set up the inequality: \[ \frac{360}{n} + 30 - 0.25n < 39 \]

Step 5: Solve the Inequality

Rearrange the inequality: \[ \frac{360}{n} + 30 - 0.25n < 39 \] Simplify to: \[ \frac{360}{n} - 0.25n < 9 \] Multiply the entire inequality by \(n\) to eliminate the fraction and solve for \(n\): \[ 360 - 0.25n^2 < 9n \] Rearrange this into a quadratic inequality: \[ 0.25n^2 + 9n - 360 > 0 \] Multiply through by 4 for convenience: \[ n^2 + 36n - 1440 > 0 \] Use the quadratic formula to solve for \(n\): \[ n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] where \(a = 1\), \(b = 36\), and \(c = -1440\).

Step 6: Quadratic Solution

Calculate the discriminant: \[ b^2 - 4ac = 36^2 - 4(1)(-1440) = 1296 + 5760 = 7056 \] Find the roots: \[ n = \frac{-36 \pm \sqrt{7056}}{2} \] \[ \sqrt{7056} = 84 \] Thus, \[ n = \frac{-36 \pm 84}{2} \] The solutions are: \[ n = \frac{48}{2} = 24 \] and \[ n = \frac{-120}{2} = -60 \]

Step 7: Interpret the Solution

Since \(n\) must be a positive integer, ignore \(n = -60\). Thus, \(n = 24\) indicates where the inequality flips from negative to positive. To ensure costs per person are less than \$39, \(n\) must be greater than 24.

Key Concepts

Quadratic Equations

Quadratic equations are essential in solving problems where the relationship between variables is not merely linear but involves squares. These equations generally take the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( x \) represents the variable you are solving for.
This type of equation can often have two solutions due to the squared term, which is why it is so handy in scenarios involving inequalities and optimization.
In the context of cost analysis, quadratic equations help determine points where certain conditions are met, such as minimizing costs or maximizing value. The step-by-step solution to the riverboat theater problem included solving a quadratic inequality to find the minimum group size.

• The discriminant \( b^2 - 4ac \) tells us how many real solutions exist. Positive results indicate two real solutions, zero means one solution, and negative implies no real solutions.
• The quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) provides an efficient way to find these solutions.

The importance of understanding quadratic equations lies in their ability to model real-world situations, making them extremely practical in various fields, including cost analysis and more.

Cost Analysis

Cost analysis involves systematically evaluating all factors that determine the cost of a product or service. In the riverboat theater scenario, calculating costs is crucial not just for participants but also to ensure the journey remains affordable when factoring in the group discount.
This involves understanding two main components of the cost:

• **Fixed Costs** like the bus hire which stays constant irrespective of the number of people. For this exercise, dividing the fixed cost by the number of group members helps establish per person expense, encouraging larger groups to reduce individual costs.


• **Variable Costs** such as discounted theater tickets, which fluctuate with the number of people. The more group members, the greater the discount per ticket, showing a clear incentive to expand the group size to reduce costs further.

Cost analysis helps in decision-making by evaluating whether the overall structure and discounts are beneficial. In practical applications, understanding such cost elements supports better budgeting and informed financial decisions.

Discount Calculation

Discount calculation can significantly impact overall cost, especially in group scenarios like the riverboat theater problem. Understanding the way discounts work is key to optimizing expenses.
Here's how discount calculations were applied in the exercise:

• The base price of theater tickets was \$30 with a discount mechanism based on group size. The discount per ticket was \(0.25 \times n\), where \( n \) is the number of people.


• This method leads to progressively larger discounts as the group size increases, encouraging greater participation.


• The equation for the discounted ticket price per person was \(30 - 0.25n\), effectively reducing the cost as \( n \) rises.

Such discount strategies aren't just limited to this exercise; they are commonly employed in retail and services to boost sales and engage more customers. Understanding how discounts apply and affect costs is crucial for savvy financial planning and maximizing savings.