Problem 36 - Algebra 1: Concepts and Skills

Question

Write an exponential decay model for the situation. Then graph the model and use the graph to estimate the value at the end of the given time period. A population of 2,000,000 decreases by 2% per year for 15 years.

Step-by-Step Solution

Answer

The exponential decay model for the given situation is \( P(t) = 2000000(0.98)^t \). Using a graph or direct substitution, the estimate would be the value of \( P(t) \) at \( t = 15 \) years.

Step 1: Identify the Initial Quantity, Rate of Decrease, and Time Period

The initial population is 2,000,000. The rate of decrease is 2% per year. The time period is 15 years.

Step 2: Formulate the Exponential Decay Model

We can represent the exponential decay model with the equation \( P(t) = P_0 (1 - r)^t \), where \( P(t) \) is the population at year t, \( P_0 \) is the initial population, r is the annual decrease rate in decimal form, and \( t \) is the time in years. For this problem, \( P(t) = 2000000(1 - 0.02)^t \).

Step 3: Graph the Model

Plot the above function on a graph, with time (t) on the x-axis and population size \( P(t) \) on the y-axis. Because the population is decreasing, the graph would be a decreasing curve.

Step 4: Estimate the Value At the End of the Time Period

Using the graph, estimate the value of \( P(t) \) when \( t = 15 \) years. If a more accurate value is needed, substitute \( t = 15 \) in the formulated equation to find the exact population.

Key Concepts

Mathematical Modeling

Mathematical modeling is like creating a blueprint that shows how different things change over time. In our problem, we're modeling how a population of 2 million people shrinks by 2% each year using exponential decay. This is a special kind of model that's perfect for describing how populations decrease steadily over time.

To create a mathematical model, we use formulas and equations. In this exercise, the formula is for exponential decay: \[ P(t) = P_0 (1 - r)^t \]

• \( P_0 \) (Initial Population): 2,000,000
• \( r \) (Decay Rate): 0.02 (or 2% in decimal form)
• \( t \) (Time in Years): 15

This formula allows us to calculate what the population will be at any given time \( t \). By understanding how to set up this model, we can predict future sizes under similar conditions.

Population Dynamics

Population dynamics describes how populations of living things, like humans, animals, or even bacteria, change over time. It involves births, deaths, and how these affect the overall number of individuals. In this problem, we're focusing on a population that shrinks at a consistent rate.

The decrease by 2% annually is an example of a specific type of population change. This shrinking isn't random—it occurs in a pattern that can be predicted and analyzed. By using mathematical modeling, scientists and researchers can make forecasts about populations. These predictions help in planning, conservation, and understanding environmental impacts.

• Models help identify future population sizes.
• Can be used in urban planning, resource management, etc.

Population dynamics give us the tools to study and manage the growth or decline of groups over time.

Graphing Exponential Functions

Graphing exponential functions is a powerful way to visually understand how quantities increase or decrease. In this case, we want to graph the exponential decay of a population.

When we graph our function, \( P(t) = 2000000(1 - 0.02)^t \), we place time \( t \) on the x-axis and population \( P(t) \) on the y-axis. As time progresses, the curve slopes downwards, showing a steady decrease.

• Each point on the graph represents the population at one year.
• At \( t = 15 \), we can estimate the population value by looking at the corresponding point on the graph.

Plotting these points helps us see how fast the population shrinks and reveals the long-term effects of the 2% annual decrease. Even if we don't calculate each step by hand, the graph provides a quick visual summary of the trends in population size over the 15-year span.