Problem 26 - College Algebra and Calculus: An Applied Approach

Question

Find three positive numbers \(x, y\), and \(z\) that satisfy the given conditions. The sum is 120 and the sum of the squares is minimum.

Step-by-Step Solution

Answer

The obtained solution verifies that each of the numbers \(x, y, z\) is 40. This implies that \(x^2 + y^2 + z^2\) is minimum when each of the numbers is 40.

Step 1: Set up the equations

Let's denote the three numbers as \(x\), \(y\), and \(z\). According to the problem, these three numbers have to satisfy the following conditions: \[x + y + z = 120\] and we need to minimize the sum of their squares \[f(x, y, z) = x^2 + y^2 + z^2 \]

Step 2: Express two variables in terms of the third one

According to the first equation, it's possible to express \(y\) and \(z\) in terms of \(x\). This will simplify the problem to an unconstrained optimization problem. From \[x + y + z = 120\], we can express \(y\) and \(z\) in terms of \(x\), namely, \(y = 120 - x - z\) and plug this into \(f(x, y, z)\).

Step 3: Differentiate and set the derivative equal to zero

After substituting \(y\) into our function, differentiate the function \(f(x, y, z)\) with respect to the variable \(x\) and set the derivative equal to zero. This will provide the optimized value of \(x\) that yields the minimum value.

Step 4: Solve for \(x\), \(y\), \(z\)

After finding the value for \(x\), substitute this value back into the first equation to solve for other variables \(y\) and \(z\)

Step 5: Validate the solution

A final verification step is essential to be certain that the obtained solution indeed provides a minimum.

Key Concepts

Sum of squares minimization

In algebra, optimization often involves finding the minimum or maximum value of a particular function. One common problem is minimizing the sum of squares. For the problem at hand, we aim to minimize the sum of the squares of three variables, given by the function \(f(x, y, z) = x^2 + y^2 + z^2\). This serves as our objective function which needs to be minimized. This type of problem allows us to find the closest sum while balancing the input variables. In practical terms, minimizing the sum of squares can lead to less variability in measurements or observations.

The expression \(x^2 + y^2 + z^2\) inherently seeks to ensure each component contributes less to the overall total, leading to a more equal or balanced distribution among the variables. This approach is applicable in statistical models and variance minimization techniques where maintaining equilibrium among parameters is crucial. The solution to these problems often leads to scenarios where extremes are avoided and the values are distributed evenly.

Positive numbers constraint

Constraints are conditions that the solutions to an optimization problem must satisfy. In this situation, one of the key constraints is that the values of \(x\), \(y\), and \(z\) must be positive numbers. This adds an additional layer of complexity because not only do we need a solution that minimizes the sum of squares, we also need each solution value to be greater than zero.

When dealing with positive number constraints, remember the basic rules of inequalities. It ensures that no value of the variables dips below zero, maintaining a realistic and logical scenario.

• For instance, \(x + y + z = 120\) ensures that when one variable increases, others must decrease to maintain the sum at 120.
• This interplay among the variables while being bound by positivity is essential in many fields such as economics, logistics, and resource management, where negative values have no feasible implication.

Positive constraints maintain the integrity of the problem’s real-world applicability, ensuring outcomes are tangible and applicable.

Constrained optimization

Constrained optimization is an essential concept in mathematics where optimization occurs with respect to given constraints or limitations. In our exercise, we deal with both an equality constraint \(x + y + z = 120\) and inequality constraints ensuring that \(x, y, z\) are positive.

The method typically applied is called the method of Lagrange multipliers, which ingeniously allows us to consider the constraint as part of the problem, finding a local minimum or maximum of a function subject to an equality constraint. While this exercise simplifies a bit by replacing some variables using the constraint to reduce dimensions, the key principle remains the same: Find optimal parameters that fit the given restrictions.

Important steps include:

• Substituting one or more variables via the equality constraint to simplify the objective function.
• Using derivatives to find critical points, verifying if these points indeed provide a minimum (or maximum).

By integrating constraints directly into the approach, constrained optimization ensures solutions are not just mathematically efficient but also adhere to all required real-world limits and rules.