Problem 52 - Thomas Calculus in SI Units

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 the Variables

We want to determine the number of people, let's call this variable \( x \), that would maximize our profit. The number of people booked for the tour is at least 50 and at most 80.

Step 2: Establish the Revenue Function

The revenue function \( R(x) \) is calculated as the number of people \( x \) multiplied by the price per person. The price per person decreases by \$2 for each person over 50. This means the price is \( 200 - 2(x - 50) = 200 - 2x + 100 = 300 - 2x \). Therefore, the revenue function is \( R(x) = x \times (300 - 2x) = 300x - 2x^2 \).

Step 3: Calculate the Cost Function

The cost function \( C(x) \) includes the fixed cost of \\(6000 plus \\)32 per person. So, \( C(x) = 6000 + 32x \).

Step 4: Establish the Profit Function

Profit \( P(x) \) is the difference between revenue and cost. Thus, \( P(x) = R(x) - C(x) = (300x - 2x^2) - (6000 + 32x) \). Simplifying, \( P(x) = -2x^2 + 268x - 6000 \).

Step 5: Find the Maximum Profit Using Quadratic Formula

Since the profit function is quadratic, a maximum is found at its vertex. The vertex formula for \( ax^2 + bx + c \) is \( x = -\frac{b}{2a} \). Here, \( a = -2 \) and \( b = 268 \). Therefore, the number of people that maximizes profit is \( x = -\frac{268}{2 \times -2} = \frac{268}{4} = 67 \).

Step 6: Verify Constraints

We check if our result of 67 satisfies the constraints of at least 50 and at most 80 people. Since 67 lies within this range, it is a valid solution.

Key Concepts

Revenue Function

The revenue function is an essential part of understanding profit maximization for any business. It represents the total income generated by the sale of goods or services. For our tour service scenario, the revenue function takes into account the number of people joining the tour and the price per person.

In this case, the revenue function, denoted as \( R(x) \), is determined by multiplying the number of participants \( x \) by the adjusted price per person. We start with a base price of \( \\( 200 \) per person, which decreases by \( \\) 2 \) for each additional person over the minimum requirement of 50 participants. So the actual price becomes \( 300 - 2x \). This leads to the revenue function:

• \( R(x) = x \times (300 - 2x) = 300x - 2x^2 \)

The decreasing price reflects in the equation as a negative \( x^2 \) term, showcasing how revenue changes with varying group sizes.

Cost Function

Next, let's break down the cost function. It represents the total costs incurred to offer the service. Costs in our problem are split into fixed and variable costs.

• The fixed cost in this scenario is \( \\( 6000 \), covering general expenses needed irrespective of the number of tour attendees.
• The variable cost is \( \\) 32 \) per person, representing additional costs incurred as more people join the tour, like food, transportation, etc.

Therefore, the total cost function, denoted as \( C(x) \), is:

• \( C(x) = 6000 + 32x \)

This function is linear with respect to \( x \), meaning costs increase uniformly with each additional participant.

Quadratic Formula

The quadratic formula is a vital tool for solving quadratic equations of the form \( ax^2 + bx + c = 0 \), which often appear in profit analysis as seen in our problem's profit function.

The formula is:

• \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

In our scenario, the profit function \( P(x) = -2x^2 + 268x - 6000 \) fits the general quadratic equation format, with \( a = -2 \), \( b = 268 \), and \( c = -6000 \). While calculating profit maxima, the vertex formula, which is derived from the quadratic formula, directly helps in finding the peak of the parabola.

Vertex of a Quadratic

The vertex of a quadratic function provides crucial information, especially when it comes to maximization or minimization problems. It signifies the turning point of the parabola represented by the quadratic equation.

• The vertex can be calculated by the formula \( x = -\frac{b}{2a} \), where \( a \) and \( b \) are coefficients from the quadratic equation \( ax^2 + bx + c \).

In the context of our profit function \( P(x) = -2x^2 + 268x - 6000 \), we substitute \( a = -2 \) and \( b = 268 \) into the formula:

• \( x = -\frac{268}{2 \times -2} = \frac{268}{4} = 67 \)

This result indicates that 67 people on the tour result in the maximum profit, confirming the vertex's role in determining optimal solutions within given constraints.