Problem 44 - Thomas Calculus

Question

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

Step-by-Step Solution

Answer

67 people maximize the profit.

Step 1: Define Variables

Let \( x \) represent the number of additional people over 50. Therefore, the total number of people is \( 50 + x \). Each additional person decreases the rate by \$ 2, so the rate per person becomes \( 200 - 2x \).

Step 2: Model Revenue Function

The revenue \( R \) is calculated by multiplying the number of people by the rate per person. Thus, \( R = (50 + x)(200 - 2x) \). Simplify this function to \( R = 10000 - 100x - 2x^2 + 200x \), which simplifies further to \( R = 10000 + 100x - 2x^2 \).

Step 3: Model Cost Function

The cost \( C \) comprises a fixed cost of \\( 6000 and \\) 32 per person. Hence, cost is given by \( C = 6000 + 32(50 + x) \). This simplifies to \( C = 7600 + 32x \).

Step 4: Profit Function

Profit \( P \) is calculated as \( P = R - C \). Substitute the equations from Step 2 and Step 3: \( P = (10000 + 100x - 2x^2) - (7600 + 32x) \). Simplify to get \( P = 2400 + 68x - 2x^2 \).

Step 5: Determine Maximum Profit

The profit function \( P = 2400 + 68x - 2x^2 \) is a quadratic function. To find the maximum, use the vertex formula for a parabola \( x = -\frac{b}{2a} \). Here, \( a = -2 \) and \( b = 68 \). Substitute to get \( x = -\frac{68}{2(-2)} = 17 \).

Step 6: Calculate Total Number of People

Remember that \( x \) represents the number of additional people. Thus, the total number of people is \( 50 + x = 50 + 17 = 67 \).

Key Concepts

Understanding Quadratic Functions

Quadratic functions play a vital role in many areas of economics, especially when it comes to profit maximization scenarios. A quadratic function is a polynomial of degree two, and it takes the general form: \[ f(x) = ax^2 + bx + c \]where \(a\), \(b\), and \(c\) are constants, and \(a eq 0\). The graph of a quadratic function is a parabola, and it can open upwards or downwards depending on the sign of \(a\). If \(a\) is positive, the parabola opens upwards, and it has a minimum point. Conversely, if \(a\) is negative, the parabola opens downwards, like in our tour service problem, and it has a maximum point.
Quadratic functions are often used in profit scenarios because they can model situations where increases in one variable, after a certain point, lead to decreases in another. In our problem, increasing the number of tourists initially increases profit, but past a certain point (the vertex), the cost outweighs the benefits, leading to decreased profit.

The Essentials of Cost Analysis

Cost analysis is an essential component of understanding a business’s efficiency and potential profitability. In the context of the tour service, the costs were broken down into fixed and variable components. Fixed costs are expenses that do not change with the level of output or sales. In our scenario, the fixed cost is \(\\(6000\), no matter how many additional people join the tour.
Variable costs, on the other hand, are costs that vary with production volume; here, it's represented by \(\\)32\) per person. The cost function given was:\[ C = 6000 + 32(50 + x) \]This formula captures how both fixed and variable costs contribute to overall expenses. Cost analysis not only helps in determining break-even points but also plays a critical role in pricing and evaluating strategies for profit maximization.

Applying the Vertex Formula

The vertex formula is an essential tool when working with quadratic functions, especially in finding the maximum or minimum values of a parabolic graph. In business terms, like in our profit maximization problem, the vertex of the parabola represents the optimal point — the maximum profit in this context.
The vertex can be found using the formula:\[ x = -\frac{b}{2a} \]For our profit function \( P = 2400 + 68x - 2x^2 \), we identified that \(a = -2\) and \(b = 68\). Plugging these values into the formula gives:\[ x = -\frac{68}{2(-2)} = 17 \]This result tells us that adding 17 more people over the minimum of 50 maximizes profit. Thus, the total number of people that should be on the tour for maximum profit is \(50 + 17 = 67\). Using the vertex formula allows us to efficiently find the point of maximum profitability without evaluating each possible number of participants individually.