Problem 57 - Algebra 1

Question

In \(1990,\) a company had a profit of \(\$ 173,000,000 .\) In \(1996,\) the profit was \(\$ 206,000,000 .\) If the profit increased the same amount each year, find the rate of change of the company's profit in dollars per year.

Step-by-Step Solution

Answer

The rate of change of the company's profit from the year 1990 to 1996 is \$5,500,000 per year.

Step 1: Identify The Given Parameters

The initial year is 1990 with an initial profit of \$173,000,000. The final year is 1996 with a final profit of \$206,000,000. The change in profit is the final profit minus the initial profit, which is \$206,000,000 - \$173,000,000 = \$33,000,000. The change in time is the final year minus the initial year, or 1996 - 1990 = 6 years.

Step 2: Calculate The Rate Of Change

Apply the rate of change formula, which is the change in profit divided by the change in time. In this case, it will be \$33,000,000 / 6 years.

Step 3: Simplify The Result

By doing the calculation, the rate of change in profit per year is approximately \$5,500,000 per year.

Key Concepts

Algebraic Concepts

Algebra is a branch of mathematics that deals with symbols and the rules for manipulating those symbols; it's a language that describes patterns, relationships, and changes. One key algebraic concept is the rate of change, which describes how one quantity changes in relation to another. In the context of our example, the profit of a company changed over a certain period.

In order to understand the company's performance over time, algebra helps us create an equation that expresses the profit increase per year. When we talk about the rate of change, we are talking about the speed at which the company's profit is increasing (or decreasing), and we calculate it by comparing the change in profit to the change in time.

Linear Functions

A linear function describes a constant rate of change, resulting in a straight line when plotted on a graph. The relationship between two variables can be depicted by a linear equation of the form y = mx + b, where y represents the dependent variable, x represents the independent variable, m is the slope, and b is the y-intercept. In our exercise example, if we consider the profit of the company as y and the number of years as x, we can utilize these values to formulate a linear equation representing the company's profit over time.

Assuming the profit increases consistently, pinpointing any year's profit is straightforward with this function. This is vital because it assists in enlightening us about the expected financial growth in the upcoming years, granted that the growth pattern persists unaltered.

Slope Calculation

The slope of a line in a graph measuring change is a number that describes both the direction and the steepness of the line. To calculate the slope, we often use the formula \( m = \frac{\text{change in } y}{\text{change in } x} \) where m denotes the slope. In other words, it's the rise over the run, or the vertical change over the horizontal change between two points on the line.

In the exercise's scenario, the slope represents the annual rate of change in profit, which can be computed as \( \frac{\text{\textdollar}33,000,000}{6 \text{ years}} \) or \( \text{\textdollar}5,500,000 \) per year. This tells us that, on average, the company's profit went up by this amount each year. Knowing how to calculate the slope allows students to quantify trends and make predictions based on past data, an essential skill in many real-world applications such as economics, science, and engineering.