Problem 24 - Algebra Form and Function

Question

Let \(s(t)\) give the number of acres of wetlands in a state in year t. Assuming that the area of wetlands goes down over time, say what the statements tell you about the wetlands. \(\frac{s(20)-s(10)}{20-10}=-520\)

Step-by-Step Solution

Answer

Answer: The average rate of decrease in the area of wetlands per year is 520 acres, which indicates that the wetlands are shrinking and losing area over time, possibly due to human activities, natural causes, or a combination of both.

Step 1: Identify the given information

The given information is a relationship between the areas of wetlands at years 10 and 20. The relationship is: \(\frac{s(20)-s(10)}{20-10}=-520\).

Step 2: Interpret the fraction

The fraction \(\frac{s(20)-s(10)}{20-10}\) represents the average rate of change of the area of wetlands, which indicates how the area decreases over time. Since the numerator represents the change in area and the denominator the change in time, the fraction represents the average rate of change per year.

Step 3: Interpret the sign

The fact that the average rate of change is negative (-520) indicates that the area of wetlands is decreasing over the given period from year 10 to year 20. If the area was increasing, the value would have been positive.

Step 4: Find the average rate of decrease per year

By simplifying the given expression, we can find the average rate of decrease in the area of wetlands per year: \(\frac{s(20)-s(10)}{20-10}=-520\) translates to \(\frac{\Delta s}{10}=-520\). To find the change in area per year, multiply both sides by 10: \(\Delta s=-520\times 10 = -5200\).

Step 5: Interpret the result

The result \(\Delta s=-5200\) indicates that over the 10-year period between year 10 and year 20, the area of wetlands in the state has decreased by 5,200 acres. The average rate of decrease per year is 520 acres per year. This tells us that the wetlands are shrinking, possibly due to human activities, natural causes, or a combination of both.

Key Concepts

Interpretation of Linear Functions

Linear functions represent relationships where the change is constant, often modeled by a straight line when graphed. In real-life scenarios, they describe situations where one quantity depends linearly on another. For instance, in the context of wetlands, the function \(s(t)\) relates the year \(t\) to the number of acres of wetlands.
Linear functions are often characterized by their slope, which indicates the rate at which one variable changes in response to another. The formula for the average rate of change, such as \(\frac{s(20)-s(10)}{20-10}\), is a tool used to measure how one quantity varies over a set period.
This particular linear relationship shows the average decrease in wetlands over a decade. If the change is consistent, linear models can be very accurate. However, they also assume that the rate of change does not fluctuate, something to bear in mind.

Decrease in Function Values

The decrease in function values is a critical concept in understanding how quantities reduce over time or circumstances. When a function's average rate of change, such as \(-520\), is negative, it indicates a decline in the quantity it measures.
In the example of wetlands, this negative change shows that the number of wetlands is reducing every year. The calculated average rate of decrease is \(520\) acres per year. To visualize it, imagine losing \(520\) football field-sized areas of wetlands every year.
Identifying a negative slope helps us pinpoint the factors contributing to this decline, like urban development or climate changes, and understand the speed of these changes. This knowledge is pivotal when making decisions about conservation efforts or urban planning.

Real-World Applications of Algebra

Algebra provides invaluable tools for solving real-world problems by modeling situations with mathematical expressions. These models help us predict and analyze trends and outcomes.
In the case of decreasing wetlands, algebra aids in visualizing and calculating their decline. Policymakers and environmentalists use such analyses to understand and act upon ecological issues. For example, understanding the rate of decline helps allocate resources more effectively to conservation programs.
Outside environmental science, algebra models similar trends in economics, engineering, and social sciences. For example, calculating depreciation rates of assets or predicting future population changes. By using algebra, we can tackle various challenges, making it an essential part of many fields.