Problem 318 - Calculus Volume 1

Question

Set up and evaluate each optimization problem. Find two positive integers such that their sum is 10 , and minimize and maximize the sum of their squares.

Step-by-Step Solution

Answer

Solution Steps

Step 1: Understand the Problem

We are given two positive integers whose sum is 10. We need to minimize and maximize the sum of their squares. Let's denote these integers as \( x \) and \( y \). Thus, \( x + y = 10 \). We want to find expressions for minimizing and maximizing \( x^2 + y^2 \).

Step 2: Express the Problem Mathematically

Use the equation \( x + y = 10 \) to express \( y \) in terms of \( x \): \( y = 10 - x \). Substitute this into the expression for the sum of squares: \( x^2 + y^2 = x^2 + (10 - x)^2 \).

Key Concepts

Positive Integers

When we talk about positive integers in mathematics, we refer to whole numbers greater than zero. These numbers are used in everyday counting and ordering. In the context of optimization problems, these positive integers act as our variables.
For example, if you are asked to find two numbers whose properties need examining, it's important to remember that numbers like -1, 0, or 2.5 wouldn’t qualify in this discussion.
This means that when solving our problem, we only consider numbers like 1, 2, 3, and so on.
This specific constraint simplifies the process as the range of possible solutions is limited to positive whole numbers.

Sum of Squares

The sum of squares is a fundamental concept often seen in algebra and statistics. When we say "the sum of squares," we mean taking each number in our set, squaring them, and then adding these squares together.
For two numbers, say \( x \) and \( y \), the sum of their squares is given by the formula \( x^2 + y^2 \).
This expression becomes essential in optimization problems where you aim to either minimize or maximize the result. Squaring a number makes it non-negative and often amplifies the value, thus manipulating how we interpret or solve mathematical problems.

• Squaring ensures all values are non-negative.
• It’s used to emphasize the differences or importance of certain values.

Understanding this concept is integral to solving algebraic expressions involving squares of numbers.

Algebraic Expressions

Mathematics uses algebraic expressions to represent relationships between variables in the form of mathematical statements.
These expressions often include constants, variables, and operations (like addition or subtraction).
In the optimization problem at hand, the algebraic expression \( x + y = 10 \) helps establish a relationship between two variables.
By substituting these algebraic expressions, you simplify and modify them for solving, such as rewriting \( y \) as \( 10 - x \) and substituting into \( x^2 + y^2 \).Key Uses:

• Allow expression manipulation to simplify calculations.
• Help in framing real-world problems into solvable mathematical formats.

Algebraic expressions form the skeleton of solving optimizations, setting up problems symbolically.

Minimization and Maximization

Minimization and maximization are key strategies in optimization, a significant branch of mathematics. The aim is to either find the smallest or largest value a function can take.
In this exercise, our goal was to both minimize and maximize the sum of squares of two integers whose sum is a fixed number.Process Steps:

• Identify the function to be optimized, here \( x^2 + (10-x)^2 \).
• Use algebraic manipulation or calculus to find optimal points.

The task can involve calculus, graphing, or strategic substitution to find solutions. It tests your ability to transform a word problem into a mathematical optimization challenge, making minimization and maximization crucial to solving these tasks efficiently.