Problem 74 - College Algebra and Calculus: An Applied Approach

Question

Annual Sales The annual sales \(S\) (in billions of dollars) of Microsoft Corporation from 1996 to 2006 can be approximated by the linear equation \(S=3.54 t-13.1, \quad 6 \leq t \leq 16\) where \(t\) represents the year, with \(t=6\) corresponding to 1996\. Use the model to estimate the year in which Microsoft's annual sales were about \(\$ 20,000,000,000\).

Step-by-Step Solution

Answer

Microsoft's annual sales were estimated to be about $20 billion in the year 2000.

Step 1: Understanding the given values

The sales \(S\) is given as $20 billion, which needs to be converted to the same scale used in the equation, i.e., billions of dollars. So, the \(S\) to be substituted in the given linear equation is 20 (not 20,000,000,000).

Step 2: Substitute \(S\) into the equation

Replace \(S\) with 20 in the equation \(S=3.54t-13.1\), obtaining the equation \(20 = 3.54t - 13.1\). This equation is to be solved for \(t\).

Step 3: Solving for \(t\)

Transform the equation by adding 13.1 to both sides, getting \(20 + 13.1 = 3.54t\), which further simplifies to \(33.1 = 3.54t\). Divide both sides by 3.54 to solve for \(t\), obtaining \(t = 33.1/3.54\).

Step 4: Calculating the sales year

Calculate the right-hand side of \(t = 33.1/3.54\), which gives \(t \approx 9.35\). The \(t\) value obtained represents the number of years since 1990. Add the result to 1990, and obtain the estimated sales year of approximately 1999.35. As we can only have whole years, round up to the next whole number, obtaining 2000.

Key Concepts

Understanding Linear Equations

A linear equation is one of the fundamental concepts in algebra. Its general form is \(y = mx + b\), where \(m\) and \(b\) are constants, \(y\) is the dependent variable, and \(x\) is the independent variable. The constant \(m\) represents the slope of the line, indicating the rate of change of \(y\) with respect to \(x\), while \(b\) is the y-intercept, which is the value of \(y\) when \(x\) is zero.

In our specific example from the textbook, the equation \(S=3.54t-13.1\) represents Microsoft's annual sales, where \(S\) is in billions of dollars, and \(t\) stands for the time in years. Since this is a linear equation, we can understand that sales are changing at a constant rate annually. The coefficient \(3.54\) reflects this rate, and \(13.1\) is analogous to a 'starting point', which adjusts the scale.

When working with linear equations, it's essential to recognize these components, as they help us grasp the rate at which changes are happening, and where they start. In real-world scenarios, this can be used to predict future events, calculate past values, or compare rates of change.

Solving for Variables

Solving for variables is the process of finding the value of unknowns that make a given equation true. This is typically done by manipulating the equation using arithmetic operations to isolate the variable of interest. Keeping our focus on the linear equation \(S = 3.54t - 13.1\), to solve for \(t\), we need to isolate it on one side of the equation.

The steps to solve for \(t\) are demonstrated in the problem's solution. First, we substitute the known value of \(S\) then perform arithmetic operations like addition or division to both sides of the equation whilst maintaining equality. When solving such equations, remember that whatever you do to one side, you must do to the other to keep the equation balanced.

By practicing these steps, students enhance their problem-solving skills and their understanding of algebraic operations. It's important to be methodical and to double-check each step for mistakes, as a small error can lead to an incorrect solution.

Mathematical Modeling with Linear Equations

Mathematical modeling involves creating a mathematical representation of a real-world scenario to analyze and make predictions. Linear equations are often used to model relationships between two variables when one variable is a constant multiple of the other, plus a fixed amount.

In the context of the Microsoft annual sales problem, the linear equation models the relationship between the year and the sales figures. Here, \(S=3.54t-13.1\) acts as our mathematical model, allowing us to estimate past or future sales as long as the linear relationship remains valid. This modeling is omnipresent in various fields, like finance, science, and engineering, helping us to understand trends and conduct scenario analysis.

When applying a mathematical model, it’s crucial to know its limitations. Models are simplifications of reality, so while they can be incredibly useful for predictions, they may not account for unexpected variables or radical changes outside the scope of the model's parameters. Understanding the assumptions and constraints of a model is just as important as being able to use it for calculations.