Problem 44 - Calculus Early Transcendentals

Question

Find three positive numbers whose sum is 12 and the sum of whose squares is as small as possible.

Step-by-Step Solution

Answer

The three numbers are 4, 4, and 4.

Step 1: Understanding the Problem

We need to find three positive numbers, say \( x \), \( y \), and \( z \), such that their sum \( x + y + z \) is 12. Additionally, the sum of their squares \( x^2 + y^2 + z^2 \) is as small as possible.

Step 2: Setting Up the Equations

From the problem, we have the equation: \( x + y + z = 12 \). We aim to minimize the expression \( x^2 + y^2 + z^2 \).

Step 3: Expressing Variables

Choose one variable to express in terms of the others using the equation \( x + y + z = 12 \). Let's express \( z = 12 - x - y \). This will allow us to substitute \( z \) in our expression for minimization.

Step 4: Substituting Expression

Substitute \( z = 12 - x - y \) into the sum of squares: \[ x^2 + y^2 + (12 - x - y)^2 \]. This expression now only involves \( x \) and \( y \).

Step 5: Simplifying and Calculating Derivative

Expand and simplify the expression: \[ x^2 + y^2 + (12 - x - y)^2 = x^2 + y^2 + 144 - 24x - 24y + 2xy + x^2 + y^2 = 2x^2 + 2y^2 + 2xy - 24x - 24y + 144 \].To find the minimum, we'll use derivatives. The partial derivatives with respect to \( x \) and \( y \) are:\[ \frac{\partial}{\partial x}(2x^2 + 2y^2 + 144 - 24x - 24y + 2xy) = 4x + 2y - 24 \]\[ \frac{\partial}{\partial y}(2x^2 + 2y^2 + 144 - 24x - 24y + 2xy) = 4y + 2x - 24 \].

Step 6: Setting Derivatives to Zero

Set the partial derivatives equal to zero:\[ 4x + 2y - 24 = 0 \]\[ 4y + 2x - 24 = 0 \].Solve these equations simultaneously to find \( x \) and \( y \).

Step 7: Solving the Equations

From the equations, solve for \( x \) and \( y \):Subtract the second equation from the first:\[ (4x + 2y) - (2x + 4y) = 24 - 24 \2x - 2y = 0 \x = y \].Substitute \( x = y \) into \( x + y + z = 12 \) gives:\[ x + x + (12 - 2x) = 12 \2x = 12 - x \3x = 12 \x = 4 \].Thus, \( x = y = 4 \), so \( z = 12 - 2x = 12 - 8 = 4 \).

Step 8: Conclusion

Thus, the three positive numbers are 4, 4, and 4, making the sum of their squares as small as possible.

Key Concepts

Partial Derivatives

In optimization problems, particularly those involving multiple variables, partial derivatives are crucial tools. They help us understand how a function changes as each variable changes, while other variables are held constant. Imagine you're holding a balloon. If you push it from one side, only that side changes, while the rest stays more or less the same. This is similar to how partial derivatives work.
For the equation we're trying to minimize,

• We have the function expressing the sum of squares: \( f(x, y, z) = x^2 + y^2 + z^2 \).
• Given that we expressed \( z \) as \( 12 - x - y \), the function becomes dependent only on two variables: \( x \) and \( y \).

To find minimum or maximum values in such scenarios, we compute the partial derivatives with respect to each variable. Here, these are \( \frac{\partial f}{\partial x} \) and \( \frac{\partial f}{\partial y} \). These derivatives tell us how the function changes as we tweak \( x \) or \( y \) slightly.
By setting these partial derivatives to zero, we find the stationary points that could potentially be minimum or maximum points. This process is like finding the flattest part of a steep hill, indicating where the hill begins to level out.

Simultaneous Equations

Once we have the partial derivatives, the next step is to set them equal to zero to find potential solutions. This is where simultaneous equations come in.
In our exercise, after computing the derivatives and setting them to zero, we get:

• \( 4x + 2y - 24 = 0 \)
• \( 4y + 2x - 24 = 0 \)

Simultaneous equations involve solving these equations together. The goal is to find values of \( x \) and \( y \) that satisfy both equations. Such solutions are often found by elimination or substitution methods.
In this example:

• We notice symmetry, indicating \( x = y \). This simplifies our task.
• Substitute back into the original constraint \( x + y + z = 12 \) to solve for each variable.

Solving these equations is like solving a puzzle where pieces fit perfectly together to give a solution that makes sense with both parts considered.

Mathematical Optimization

At its core, mathematical optimization is about finding the best value—often minimum or maximum—of a function, subject to certain conditions or constraints. In our problem, we're finding three numbers whose sum is 12 and whose sum of squares is minimized.
The process involves:

• Formulating an objective function, here \( x^2 + y^2 + z^2 \), to be minimized.
• Using constraints like \( x + y + z = 12 \) to guide the solution.

Optimization problems can be seen in everyday life, from minimizing cost and maximizing efficiency to solving practical issues like diet planning or resource allocation.
By applying techniques such as setting up partial derivatives and solving simultaneous equations, mathematical optimization gives us powerful tools to find optimal solutions effectively. In essence, it's about making the best decision out of a set of available options, using mathematics as the guiding principle.