Problem 48 - College Algebra and Calculus: An Applied Approach

Question

Writing Real-Life Problems In Exercises 47-50, solve the number problem and write a real-life problem that could be represented by this verbal model. For instance, an applied problem that could be represented by Exercise 47 is as follows. The sum of the length and width of a one-story house is 100 feet. The house has 2500 square feet of floor space. What are the length and width of the house? One number is 1 more than another number. The product of the two numbers is 72 . Find the numbers.

Step-by-Step Solution

Answer

The two numbers are 8 and 9.

Step 1: Define Variables

Let the two unknown numbers be called \(x\) and \(y\). According to the problem, one number \(y\) is 1 more than the other number \(x\). So we can also say that \(y=x+1\).

Step 2: Setup equations from the problem

The problem states that the product of the two numbers is 72, so we can express this as an equation \(x*y=72\). With \(y\) being \(x+1\), substitute \(y\) in this equation and it becomes \(x*(x+1)=72\).

Step 3: Solve the Equation

To simplify, distribute the \(x\) to both terms on the right side of the equation resulting in \(x^2 + x = 72\). Then rearrange terms to equal zero, hence the equation becomes \(x^2 + x - 72 = 0\). This equation can be factored into \((x-8)(x+9) = 0\).\nTo find the roots of the equation, set each factor equal to zero and solve for \(x\): if \(x-8=0\), then \(x=8\), or if \(x+9=0\), then \(x=-9\).\nSince \(x\) is a length and therefore must be a positive quantity, discard \(x=-9\) as extraneous.

Step 4: Finding the second number

Substitute \(x=8\) into the first equation \(y=x+1\), and solve for \(y\), hence \(y=8+1=9\). Therefore, the two numbers are 8 and 9.

Key Concepts

Verbal Model

A verbal model is a way of describing a problem in words before translating it into mathematical expressions. It's important to understand the problem thoroughly before committing anything to paper. In our number problem, the verbal model describes how one number is related to another. One number is 1 more than the other number, and their product is 72.

This approach guides us in forming the equations we need for solving the problem. By stating the relationships and conditions laid out in words, we gain a clearer picture of what we are trying to solve.

Equations

Equations are mathematical statements that express the relationships outlined in our verbal model. For this number problem, we translate the information given into equations.

Firstly, designating variables, let one number be denoted as \(x\), and since the other number is 1 more than this, it becomes \(y = x + 1\).

The product, or multiplication, of these two numbers should equal 72, hence the equation \(x \, \times \, y = 72\). Substituting \(y\) for \(x + 1\) leads us to another key equation: \(x \, \times \, (x + 1) = 72\). This equation is the foundation needed to find the numbers involved.

Roots

In algebra, the term 'roots' refers to the solutions of an equation. These are the values of the variable that satisfy the equation. For the problem at hand, once we have the equation \(x^2 + x - 72 = 0\), we need to determine its roots.

This often means transforming and simplifying the equation to find possible values of \(x\). When the equation is factored into \((x - 8)(x + 9) = 0\), it "splits" the problem into two possible solutions: setting \(x - 8 = 0\) leads to \(x = 8\), and \(x + 9 = 0\) gives \(x = -9\). Since a length cannot be negative, only the positive solution, \(x = 8\), is valid for our context.

Real-Life Problem

This theoretical exercise can be directly applied to real-life scenarios, turning abstract numbers into practical insights. For instance, in a real-life setting, this problem can represent dimensions of a rectangular plot of land. Let the width be one foot more than the length and the area to be covered exactly 72 square feet.

In such a situation, knowing how to set up and solve an equation enables us to calculate necessary dimensions efficiently. This skill is crucial when determining measurements for construction projects or setting up proper gardening layouts. Translating these mathematical skills to real-world applications makes algebra a powerful tool in everyday decision-making.