Maximizing the Product of Two Numbers Given a Sum of 54

Introduction

In this article, we’ll explore how to find two positive numbers whose sum is 54 and whose product is maximized. This type of problem is common in mathematics and can be approached using multiple methods, including calculus and the use of inequalities. We'll delve into these approaches and provide a clear, step-by-step explanation.

Using Inequalities: Geometric Mean vs. Arithmetic Mean

The first method we’ll explore involves using the inequality between the geometric mean and arithmetic mean. Let's consider two positive numbers, (a) and (b), such that (a b 54).

According to the inequality between geometric and arithmetic means (AM-GM inequality), for any two positive numbers (a) and (b), the following holds true:

[sqrt{ab} leq frac{a b}{2}]

Given (a b 54), we can substitute this into the inequality:

[sqrt{ab} leq frac{54}{2} 27]

Squaring both sides of the inequality:

[ab leq 27^2 729]

To achieve the maximum product, the equality in the AM-GM inequality must hold, which occurs when (a b). Therefore:

[a b frac{54}{2} 27]

Thus, the numbers are 27 and 27, and their product is:

[27 times 27 729]

This is the maximum product for two positive numbers whose sum is 54.

Using Calculus: Derivatives and Maxima

Let's now consider the problem using calculus. Let (a) and (b) be the two numbers such that:

[a b 54]

We want to maximize the product (P ab). Express (b) in terms of (a):

[b 54 - a]

Substitute this into the product equation:

[P a(54 - a) 54a - a^2]

Now, take the derivative of (P) with respect to (a) and set it to zero:

[frac{dP}{da} 54 - 2a 0]

Solve for (a):

[2a 54 Rightarrow a 27]

Since (a b 54), we have:

[b 54 - 27 27]

Hence, the numbers are 27 and 27, and the maximum product is:

[27 times 27 729]

Using a Table and Trial and Error

For those who enjoy trial and error or visual methods, constructing a table of possible values can help. Let's create a table with values of (a) and corresponding (b), and the resulting product:

a b Product (a x b) 0 54 0 5 49 245 10 44 440 15 39 585 20 34 680 25 29 725 30 24 720 35 19 665 40 14 560 45 9 405 50 4 200

From the table, we can see that the maximum product is achieved when (a 25) and (b 29), or vice versa, resulting in a product of 725. However, for exact equality, as previously calculated, the optimal values are (a 27) and (b 27), yielding a product of 729.

Conclusion

Through the methods of inequalities, calculus, and trial and error, we have shown that the maximum product of two positive numbers whose sum is 54 is achieved when both numbers are 27. The product is 729.