Problem 54 - Algebra and Trigonometry with Analytic Geometry

Question

A travel agency offers group tours at a rate of $$\$ 60$$ per person for the first 30 participants. For larger groups - up to 90 -each person receives a $$\$ 0.50$$ discount for every participant in excess of 30 . For example, if 31 people participate, then the cost per person is $$\$ 59.50$$. Determine the size of the group that will produce the maximum amount of money for the agency.

Step-by-Step Solution

Answer

The group size that maximizes revenue is 75 participants.

Step 1: Understanding the Problem

We need to determine the group size between 30 and 90 that will maximize the total revenue for the travel agency. The cost per participant decreases by $0.50 for each participant above 30.

Step 2: Defining the Revenue Function

Let the number of participants be denoted as \( n \). The cost per person is \( 60 - 0.5(n-30) \) for \( n > 30 \) and \( n \leq 90 \). Revenue, \( R(n) \), can be expressed as: \[ R(n) = n(60 - 0.5(n-30)) \] Simplifying, \[ R(n) = n(60 - 0.5n + 15) = n(75 - 0.5n) \] Further simplifying, this becomes \[ R(n) = 75n - 0.5n^2 \]

Step 3: Finding the Maximum Revenue

The revenue function \( R(n) = -0.5n^2 + 75n \) is a quadratic function opening downwards (because the coefficient of \( n^2 \) is negative), so it achieves a maximum at its vertex. The vertex of a quadratic \( an^2 + bn + c \) is at \( n = -\frac{b}{2a} \).

Step 4: Calculating the Vertex

Substitute the coefficients into the vertex formula \( n = -\frac{b}{2a} \). Here, \( a = -0.5 \), and \( b = 75 \), so:\[ n = -\frac{75}{2(-0.5)} = \frac{75}{1} = 75 \] Therefore, the revenue is maximized when there are 75 participants.

Step 5: Conclusion

To achieve the maximum revenue for the agency, the group should consist of 75 participants.

Key Concepts

Understanding Quadratic Functions

Quadratic functions are mathematical expressions that are defined by the general formula \( ax^2 + bx + c \). Here, \( a \), \( b \), and \( c \) are constants, and \( x \) is the variable. The graph of a quadratic function is a parabola. When the coefficient \( a \) is positive, the parabola opens upwards; when \( a \) is negative, it opens downwards. Quadratic functions are useful in modeling situations where there is a maximum or minimum point, such as projectiles in physics or in this case, revenue maximization for a business.

By understanding how to manipulate and solve quadratic functions, you can determine key points like the maximum revenue in a business problem. This is because these functions can describe the relationship between different quantities, such as the number of participants and the total revenue in a travel agency scenario.

Applying the Vertex Formula

To find the maximum or minimum of a quadratic function, we use the vertex formula. The vertex of a parabola in the quadratic function \( ax^2 + bx + c \) is given by the formula \( x = -\frac{b}{2a} \). This point, known as the vertex, is where the maximum or minimum value occurs. If the parabola opens upwards, the vertex is a minimum point; if it opens downwards, as in revenue problems like this one, the vertex is a maximum point.

In the case of the travel agency's revenue problem, replacing \( a \) and \( b \) with the given values from the revenue function \( R(n) = -0.5n^2 + 75n \) helped us find that \( n = 75 \). This means that the maximum revenue is achieved when there are 75 participants.

Implementing Group Pricing Strategies

Group pricing involves setting different price points for different sizes or types of groups. This strategy is often used to attract larger groups and maximize revenue. Businesses may offer discounts as group size increases to encourage more people to participate.

In the travel agency example, the agency charges \\(60 per person for the first 30 participants. For every additional participant above 30, they reduce the price by \\)0.50 per person. This tactic is designed to encourage larger groups, and it also requires careful calculation to find the balance point where the maximum revenue is achieved. Understanding group pricing is essential for businesses that want to optimize their pricing strategies according to customer group size.

Calculating Discounts Accurately

Discount calculation is an important aspect of pricing strategies that can influence consumer behavior. In scenarios like the travel agency's, the discount increases with each additional participant over a certain threshold. Here, the discount is a linear decrease in price by \\(0.50 for every person beyond 30 participants.

To find the actual cost per person as the group size increases, one has to perform a straightforward calculation using the given discount rate. For example, for 31 participants, the discount is \\)0.50, making the cost per person \$59.50. This continues incrementally until the group reaches the maximum sustainable size of 90. By understanding how discounts are applied, businesses can better predict how price changes will affect their revenue.