Problem 56 - Thomas Calculus

Question

You operate a tour service that offers the following rates: \begin{equation} \begin{array}{l}{\$ 200 \text { per person if } 50 \text { people (the minimum number to book }} \\ {\text { the tour) go on the tour. }} \\ {\text { For each additional person, up to a maximum of } 80 \text { people }} \\ {\text { total, the rate per person is reduced by } \$ 2 .} \\ {\text { It costs } \$ 6000 \text { (a fixed cost) plus } \$ 32 \text { per person to conduct the }} \\\ {\text { tour. How many people does it take to maximize your profit? }}\end{array} \end{equation}

Step-by-Step Solution

Answer

To maximize profit, have 80 people on the tour.

Step 1: Define Variables

Let \( x \) be the number of additional people above the minimum 50 people. Therefore, the total number of people on the tour is \( 50 + x \).

Step 2: Determine Revenue

The price per person decreases by \\(2 for each additional person. Starting at \\)200 per person with 50 people, the price per person after \( x \) additional people is \( 200 - 2x \). The revenue \( R \) as a function of \( x \) is given by:\[ R(x) = (50 + x)(200 - 2x) \]

Step 3: Calculate Cost

The total cost comprises the fixed cost and the variable cost per person, which together totals:\[ C(x) = 6000 + 32(50 + x) \]

Step 4: Profit Function

Profit \( P \) is revenue minus cost. Substitute and simplify:\[P(x) = (50 + x)(200 - 2x) - (6000 + 32(50 + x)) \]\[= (50 + x)(200 - 2x) - 6000 - 1600 - 32x\]\[= 10000 + 200x - 2x^2 - 6000 - 1600 - 32x \]\[= -2x^2 + 168x + 2400\]

Step 5: Find Maximum Profit Using Vertex Formula

The profit function is a quadratic equation in the form \( P(x) = ax^2 + bx + c \) with \( a = -2 \), \( b = 168 \). The x-value of the vertex, which gives the maximum profit, can be found with:\[ x = -\frac{b}{2a} = -\frac{168}{2(-2)} = 42\]

Step 6: Compute Optimal Number of People

Since the additional people is 42, add this to the initial 50 people:\[ 50 + 42 = 92 \]

Step 7: Verify Maximum Condition

The problem states a maximum allowed number of people is 80, therefore, the practical solution within the given constraints is 80 total people.

Key Concepts

Quadratic Functions

Quadratic functions are a special type of polynomial function with the form \( ax^2 + bx + c \). These functions create a parabola when graphed, either opening upwards or downwards depending on the sign of the leading coefficient \( a \). For a tour service determining profit, such functions can appear when revenue or cost is dependent on the number of people, as presented in this exercise. In real-world scenarios like this, quadratic functions help figure out relationships that change in a nonlinear way. For instance, though you initially earn more with additional customers, costs or price reductions can limit profit growth, forming the quadratic shape. Understanding these functions aids in identifying the optimal points or strategies to maximize benefits, as further explained with vertex formulas.

Vertex Formula

The vertex formula is a handy tool when working with quadratic functions, especially in profit maximization. The vertex of a parabola gives the maximum or minimum point for the quadratic function, depending on if it opens downwards or upwards. For a quadratic function \( f(x) = ax^2 + bx + c \), the x-coordinate of the vertex is calculated using:\[ x = -\frac{b}{2a} \]This formula helps find the exact point at which the function reaches its peak value. In terms of business scenarios, like in this exercise, it reflects the number of additional people that brings maximum profit. After determining this x-value, the real number of people is adjusted by verifying practical constraints that lay within the context.

Revenue and Cost Analysis

Revenue and cost analysis forms the backbone for deciding optimal decision-making in business. Here, revenue is calculated from the price per person and the number of customers. However, with each additional customer, the price per person drops, presented in the equation \( (50 + x)(200 - 2x) \). On the other hand, costs include fixed expenses and variable components that increase with more customers, captured by \( C(x) = 6000 + 32(50 + x) \). The core idea is to plot both revenue and costs as functions and determine their interaction, ultimately solving for when revenue minus total cost - or profit - hits its highest point. Thorough analysis helps businesses like a tour service strategically plan scaling efforts while keeping an eye on profitability.

Constraints in Optimization

Constraints in optimization problems set the boundaries for feasible solutions. While mathematical models might suggest a number beyond limits, real-world constraints must be respected. This exercise exhibits such conditions with a maximum tour group size of 80 people. Though calculations found the best scenario with 92 potential customers, this overruns the allowed limit. Thus, it's essential to adjust the theoretical maximum (92 people) to fit within 80, ensuring realistic profitability measures. Understanding constraints ensures that strategic decisions align with operational capacities, regulatory limits, and market realities, preventing impractical outcomes from purely mathematical conclusions.