Problem 88 - College Algebra with Modeling and Visualization

Question

In 1988 the number of farm pollution incidents reported in England and Wales was \(4000 .\) This number had increased at a rate of 280 per year since 1979\. (Source: C. Mason, Biology of Freshwater Pollution.) (a) Find an equation \(y=m\left(x-x_{1}\right)+y_{1}\) that models these data, where \(y\) represents the number of pollution incidents during the year \(x\) (b) Estimate the number of incidents in 1975 .

Step-by-Step Solution

Answer

The equation is \(y = 280x - 552640\); approximately 360 incidents in 1975.

Step 1: Identify Known Values

Given that in 1988, the number of farm pollution incidents was \(4000\) and it increased by \(280\) each year since 1979, we recognize 1979 as the base year. Therefore, we identify the point in time as \((1988, 4000)\), and the rate of change \(m\) as \(280\).

Step 2: Use Point-Slope Form

Use the point-slope form formula \( y = m(x - x_1) + y_1 \) where \( x_1 = 1988 \), \( y_1 = 4000 \), and \( m = 280 \), to create the equation. The equation is:\[y = 280(x - 1988) + 4000.\]

Step 3: Simplify the Equation

Simplify the equation by distributing and combining like terms:\[y = 280x - 280 \times 1988 + 4000.\]Calculating the constant term:\[280 \times 1988 = 556640\]*negative sign included.Hence,\[y = 280x - 556640 + 4000 = 280x - 552640.\]Thus, the equation becomes:\[y = 280x - 552640.\]

Step 4: Estimate Incidents in 1975

To find the number of pollution incidents in 1975, substitute \(x = 1975\) into the equation derived:\[y = 280 \times 1975 - 552640.\]Calculate step by step:\[280 \times 1975 = 553000.\]Subtracting the constant:\[553000 - 552640 = 360.\]Thus, in 1975, there were approximately \(360\) pollution incidents.

Key Concepts

Point-Slope Form

The point-slope form is a way to write linear equations quickly and efficiently. It's especially useful when you know a point on the line and the slope. The formula for this is \( y = m(x - x_1) + y_1 \), where:

• \( m \) is the slope or rate of change of the line.
• \((x_1, y_1)\) is a specific point on the line.

Using this form can be advantageous because it directly incorporates the slope and a given point, making it straightforward to switch to other forms if needed. In our exercise, the point \((1988, 4000)\) and the rate of change \(280\) helped create the equation, highlighting why the point-slope form is practical for modeling real-life data.

Rate of Change

Rate of change is a fundamental concept in linear equations, defining how much a quantity increases or decreases. In a linear equation, it is symbolized by the slope \( m \). The rate of change tells us how much the dependent variable changes as the independent variable increases by one unit. In practical terms, it gives us an idea of speed or velocity in relation to the context of the problem.
For the given exercise, the rate of change \(280\) incidents per year indicates how the pollution level increases annually. Understanding this concept helps identify trends and predict future values by maintaining a steady pattern, making it essential for correctly forming equations from real-world data.

Mathematical Modeling

Mathematical modeling uses mathematical language and equations to represent and analyze real-world scenarios. It allows us to predict future behavior and trends based on known data. In the context of the exercise, we are modeling pollution incidents in England and Wales over time using a linear equation.
Creating a mathematical model involves:

• Identifying known quantities, such as past data points (e.g., the number of incidents in 1988) and rate of change (e.g., the yearly increase of 280 incidents).
• Selecting the appropriate equation form, such as the point-slope form for linear relationships.
• Constructing the equation and using it to calculate unknown values, such as estimating incidents for a previous year like 1975.

By understanding mathematical modeling, we can develop accurate predictions and strategies to manage or address real-life problems like pollution.