Problem 21 - College Algebra and Calculus: An Applied Approach

Question

You are given the 2005 value of a product and the rate at which the value is expected to change during the next 5 years. Use this information to write a linear equation that gives the dollar value of the product in terms of the year. (Let \(t=5\) represent 2005.) 2005 Value \(\quad\) Rate . $$\$ 2540 \quad \$ 125$$ increase per year

Step-by-Step Solution

Answer

The 2005 value can be represented by the linear equation \(V = 125t + 2540\). Using this equation, one can determine the product's value in the future.

Step 1: Identify the Variables

We will represent the value of the product as \(V\) and the time (year) as \(t\). Then we take \(t = 5\) as year 2005, \(V_0 = \$2540\) as the value of the product in 2005, and \(r = \$125\) as the annual increase rate.

Step 2: Formulate the Linear Equation

A linear equation of the form \(V = mt + b\) where \(m\) ir the slope (rate of change) and \(b\) is the y-intercept (initial value) can be written. Here, the rate of change for the product is the annual increase rate \(r = \$125\) and the initial value is the value of the product in 2005 \(V_0 = \$2540\). Thus, the linear equation that gives the value of the product in terms of the year becomes \(V = 125t + 2540\).

Step 3: Substituting Given Values

Substitute the given value, which is \(t = 5\) into the formulated equation, we get: \(V = 125(5) + 2540\).

Key Concepts

Understanding Slope

In the context of linear equations, the slope is a vital component. It represents the rate of change of one variable in relation to another. Specifically, it shows how much the dependent variable changes when the independent variable increases by one unit. In simpler terms, the slope indicates how steep a line is.

For our exercise, let's look at the slope as the rate of change of the product's value over time. It's given as \( \$125 \), which means that each year, the product's value increases by 125 dollars. This steady increase is constant, illustrating a uniform change in value over time.

• A positive slope, as seen here, points to an upward trend, showing an increase in value.
• A negative slope would indicate a decrease in value over time.
• A zero slope means no change in value, depicting a flat line.

In the equation \( V = 125t + 2540 \), the number 125 is the slope, helping us understand how the value each year rises consistently.

Exploring the Y-intercept

The y-intercept is the point where the line crosses the y-axis on a graph. It provides the starting point of the line when the independent variable is zero. In our exercise, it's crucial to understand what the y-intercept represents.

For the linear equation \( V = 125t + 2540 \), the y-intercept (\( b \)) is \( \$2540 \). This is the initial value of the product. It tells us how much the product was worth at the beginning, when \( t = 5 \), which we're using as the time reference for the year 2005.

• The y-intercept allows us to understand the starting conditions of a scenario or context.
• In many real-world situations, it represents the initial deposit, balance, or starting condition before any changes occur.
• The y-intercept is crucial for establishing the timeline and initial condition of the equation’s scenario.

Understanding the y-intercept helps us frame the timeline and lets us see how starting values are determined in linear equations.

The Concept of Rate of Change

Rate of change is a core aspect of linear equations and is closely related to the slope. It explains how one quantity changes in relation to another. In our specific exercise, the rate of change is the key dynamic element of the equation.

Given as \( \$125 \) per year, it indicates the amount by which the product's value increases annually. It provides a predictable, measurable way of understanding how the value grows from year to year, giving the equation practical relevance in predicting future values.

• The rate of change here is constant, meaning it doesn't accelerate or decelerate over the years.
• This consistency allows for straightforward predictions about future values of the product.
• By applying this rate of change, we effortlessly use the linear model \( V = 125t + 2540 \) to estimate values in years past 2005.

The rate of change allows us to look into the future value increases with confidence, as it remains uniform, facilitating long-term predictions easily.