Maximizing the Product of Two Numbers with a Given Difference

In this article, we explore the problem of finding two numbers that have a difference of 8 and maximize their product. We will use algebraic methods to derive the solution and understand the underlying mathematical principles.

Problem Statement

Suppose we have two numbers x and y with the following conditions:

x - y 8 Maximize the product P x cdot y

Our goal is to find the values of x and y that satisfy these conditions and maximize the product P.

Step-by-Step Solution

Expressing x in Terms of y

From the equation x - y 8, we can express x in terms of y as follows:

x y 8

Substituting into the Product Equation

Substituting x y 8 into the product equation P x cdot y, we get:

P (y 8) cdot y y^2 8y

Maximizing the Quadratic Function

The function P y^2 8y is a quadratic equation, which can be written in the form y^2 8y c. To find the maximum value, we need to determine the vertex of the parabola. The vertex form of a quadratic function is given by:

y -frac{b}{2a}

Here, a 1 and b 8. Therefore:

y -frac{8}{2 cdot 1} -4

Finding the Corresponding Value of x

Substituting y -4 back into the equation x y 8, we get:

x -4 8 4

Thus, the two numbers that have a difference of 8 and maximize their product are x 4 and y -4.

Summary: The values of x and y that maximize the product while maintaining a difference of 8 are x 4 and y -4.

Questions and Further Exploration

To ensure a comprehensive understanding, consider the following questions:

What would happen if we tried to find the coordinates on a 3-dimensional plot? Can you provide an alternative method to solve the same problem? What if the difference between the numbers is not a constant but a variable?

Feel free to explore these questions to deepen your understanding of the algebraic and calculus methods used to solve such problems.

Conclusion

In conclusion, by using algebraic methods, we have determined that the two numbers that have a difference of 8 and maximize their product are 4 and -4. This method can be applied to similar problems involving the maximization or minimization of functions with given constraints.