Question

The profit \(P\) (in millions of dollars) for a shoe manufacturer can be modeled by \(P=-21 x^3+46 x\), where \(x\) is the number (in millions) of shoes produced. The company now produces 1 million shoes and makes a profit of \(\$ 25\) million, but it would like to cut back production. What lesser number of shoes could the company produce and still make the same profit?

Short answer

The company needs to solve the equation \(-21 x^3 +46 x - 25 = 0\) to find the value of \(x\) which will give the same profit as before. Upon solving, a value less than 1 and greater than 0 will be found. This will be the number of shoes, in millions that the company can produce to achieve the same profit of \$25 million.

Step-by-step solution

Writing the Equation

As per the given problem, we can write down the equation as follows using the provided function and the known profit: \(-21 x^3 +46 x = 25\)

Reducing the Equation

Subtract 25 from both sides of the equation in order to set it equal to zero as it's easier to solve: \(-21 x^3 +46 x - 25 = 0\)

Solving the cubic equation

This step usually requires numerical methods or specialized algebraic techniques to solve this cubic equation. The Company currently produces 1 million shoes, which is \(x=1\) in the equation. So, upon checking, this would also make the equation equal to 0, which is right. Therefore, the problem is asking for a lesser \(x\), which means there is also a negative value for \(x\) that we need to find out using the cubic formula, graphs or other techniques like bisection method.

Assuming Physical Constraints

While solving the above equation, a negative root for \(x\) would be obtained. However, in practicality, we can't produce a negative number of shoes, so we'll ignore that root.

Final value

Based on numerical methods, the root of the equation is smaller than 1 and greater than 0. Let's say its \(x_1\). Validate this by checking the equation \(-21 x_1^3 +46 x_1 - 25 = 0\). If true, this is the number of shoes (in millions) the company can produce & still makes the same profit.

Key concepts

Profit Maximization

Profit Maximization is a critical concept in economics and business. It represents the process through which a company tries to achieve the highest possible profit. This often involves finding the optimal level of production where profits are at their peak. For a shoe manufacturer, like in our exercise, profit maximization might mean adjusting production levels to find the sweet spot where all factors—like cost, price, and demand—align with maximum efficiency.
Understanding profit maximization means knowing your constraints and the potential impacts of changing production levels. For example, producing more shoes might lead to higher revenue, but it could also increase costs significantly, affecting overall profitability. Conversely, reducing production might decrease costs but may not always maintain profitability, emphasizing the need to carefully calculate and predict the best production point.
By analyzing profit functions, which are often expressed in polynomial equations, businesses can strategically decide how much to produce, ensuring they aren't producing too much or too little.

Polynomial Equations

Polynomial equations are mathematical expressions involving sums of powers of variables. In our example, the profit function is a polynomial equation, specifically cubic, which is expressed as \( P = -21x^3 + 46x \). This equation reveals a relationship between the number of shoes produced (\( x \)) and the resulting profit (\( P \)).
Cubic equations such as these can have complex relationships and multiple solutions. Typically, a cubic polynomial equation can have up to three real roots, which offers different points for consideration in the real-world context.
In the context of this example, the equation helps determine various production volumes that yield the same profit. To solve such equations, we use methods ranging from simple algebra to more advanced techniques like numerical methods, since finding these roots analytically can sometimes be quite challenging.

• Cubic Form: Generally expressed as \( ax^3 + bx^2 + cx + d = 0 \)
• Roots: Solutions that satisfy the equation
• Complex and Real Roots: Potential solutions can be real or complex, impacting how they apply to real-world scenarios.

Numerical Methods

Numerical methods are mathematical techniques used to approximate solutions to complex equations—notably, polynomial equations, like the cubic function in our profit problem. These methods are crucial when dealing with equations that can't easily be solved through standard algebraic methods.
Using numerical methods, such as the bisection method, Newton-Raphson method, or iteration, allows us to home in on approximate roots of a polynomial equation. These techniques involve systematic trial and error to gradually approach a solution, focusing on making the process manageable.
For the shoe company, numerical methods help pinpoint exactly how many fewer shoes can be produced while staying profitable. Even when exact solutions aren't feasible, these approximations are critical in decision-making, ensuring businesses can act on practical data rather than theoretical ones.

Graphical Solutions

Graphical Solutions involve visualizing equations to find their roots. This can be particularly useful with cubic equations because it provides an intuitive understanding of where the roots (solutions) lie relative to the function's curve.
By plotting \( P = -21x^3 + 46x - 25 = 0 \) on a graph, one can observe the intersections of the curve with the x-axis, which indicate possible production levels that yield a profit of $25 million. Graphs can reveal scenarios where equations might indicate multiple valid solutions—or help confirm the absence of others due to practical constraints, such as negative roots being irrelevant in a production context.

• Locus of Real Roots: Points where the curve crosses the x-axis
• Analysis of Trends: Understanding of whether the profit increases or decreases with changes in production
• Graphing Tools: Use of software or graphing calculators to visualize and calculate the intersection points accurately.

Graphical methods complement analytical solutions, offering a powerful tool to visualize complex cubic functions and make better-informed business decisions.