Problem 334 - Calculus Volume 1

Question

For the following exercises, consider a pizzeria that sell pizzas for a revenue of \(R(x)=a x\) and costs \(C(x)=b+c x+d x^{2},\) where \(x\) represents the number of pizzas. Assume \(\quad\) that \(\quad R(x)=15 x, \quad\) and \(C(x)=60+3 x+\frac{1}{2} x^{2} . \quad\) How many pizzas sold maximizes the profit?

Step-by-Step Solution

Answer

Selling 12 pizzas maximizes profit.

Step 1: Define the Profit Function

The profit function is the difference between revenue and cost, denoted by \( P(x) = R(x) - C(x) \). Substitute the given functions into this equation: \( P(x) = 15x - (60 + 3x + \frac{1}{2}x^2) \).

Step 2: Simplify the Profit Function

Simplify the expression from Step 1: \( P(x) = 15x - 60 - 3x - \frac{1}{2}x^2 \). Combine like terms to get \( P(x) = -\frac{1}{2}x^2 + 12x - 60 \).

Step 3: Find the Vertex of the Quadratic Function

The profit function is a downward-opening parabola because the coefficient of \( x^2 \) is negative. The vertex of this parabola will give us the value of \( x \) that maximizes profit. The formula for the vertex \( x \) in a quadratic \( ax^2 + bx + c \) is \( x = \frac{-b}{2a} \).

Step 4: Calculate the Maximizing Value of x

Substitute the values into the vertex formula: \( x = \frac{-12}{2(-1/2)} = \frac{-12}{-1} = 12 \). Therefore, selling 12 pizzas maximizes the profit.

Key Concepts

Revenue function

In any business setting, understanding how revenue changes with the number of products sold is crucial. The revenue function, often denoted as \( R(x) \), describes how much money a business makes from selling \( x \) units of a product. In the context of our pizzeria, the revenue function is given as \( R(x) = 15x \), where \( x \) represents the number of pizzas sold. This means that for each pizza sold, the pizzeria earns \(15.

It's a linear function, which implies the revenue increases at a constant rate of \)15 per pizza. Understanding the revenue function is the first step in calculating profit, as profit is the revenue remaining after covering all costs. For businesses, knowing the revenue from each item helps in strategizing the pricing and volume sales necessary to meet financial goals.

Cost function

The cost function, \( C(x) \), represents the total cost incurred when producing \( x \) units of a product. For our pizzeria, this is shown as \( C(x) = 60 + 3x + \frac{1}{2}x^2 \). This equation is composed of three parts:

• The fixed cost of \(60, which remains constant regardless of the number of pizzas made.
• The variable cost \)3 per pizza, reflecting costs that change linearly with the number of pizzas produced.
• The quadratic term \( \frac{1}{2}x^2 \), indicating that costs increase at an increasing rate as more pizzas are produced.

This combination of costs offers insights into the expenses associated with scaling production. It's essential for determining profitability, as total costs must be subtracted from revenue to determine profit. The cost function helps businesses optimize their operations and reduce unnecessary expenses.

Quadratic equations

Quadratic equations are a pivotal part of understanding various aspects of profit maximization. A quadratic equation is typically in the form \( ax^2 + bx + c \). In our scenario, the profit function, after simplifying, is expressed as \( P(x) = -\frac{1}{2}x^2 + 12x - 60 \). This equation is a downward-opening parabola because the coefficient of \( x^2 \) is negative.

The shape and direction of the parabola signal whether the function has a maximum or minimum point. Here, the downward-opening nature of the parabola tells us there is a maximum profit point. Solving quadratic equations involves finding the roots or using the vertex formula to find the maximum or minimum values. In business, these equations help define ranges for variables that maximize profit or minimize costs.

Vertex formula

The vertex formula is vital for finding the peak or trough of a quadratic equation, which in our case concerns pinpointing the number of pizzas that maximizes profit. The formula for the vertex of a quadratic function \( ax^2 + bx + c \) is \( x = \frac{-b}{2a} \).

For the profit function \( P(x) = -\frac{1}{2}x^2 + 12x - 60 \), applying the vertex formula allows us to solve for \( x \) efficiently. By substituting \( a = -\frac{1}{2} \) and \( b = 12 \), the calculation is \( x = \frac{-12}{2(-1/2)} = 12 \). This means that selling 12 pizzas results in the highest possible profit for the pizzeria.

The vertex formula simplifies finding optimal solutions within quadratic relationships, making it a crucial tool in business for maximizing output and financial return.