Problem 58 - Calculus and its applications

Question

We have now studied models for linear, quadratic, exponential, and logistic growth. In the real world, understanding which is the most appropriate type of model for a given situation is an important skill. For each situation, identify the most appropriate type of model and explain why you chose that model. List any restrictions you would place on the domain of the function. The decrease in population of a city after its principal industry closes

Step-by-Step Solution

Answer

An exponential decay model is appropriate due to rapid initial decrease. Use for non-negative time, starting post-industry closure, until population stabilizes.

Step 1: Identify the Trend

The city’s population is expected to decrease since the principal industry has closed. This suggests a decline rather than growth.

Step 2: Consider Linear and Exponential Models

A linear model, where the decrease happens at a constant rate over time, doesn’t realistically fit situations where external factors can accelerate decline (like unemployment or migration). An exponential model, where the decrease is initially rapid and gradually slows down, might fit better in this situation.

Step 3: Evaluate a Quadratic Model

A quadratic model involves more variation in rate changes and typically includes a turning point, which doesn't align with a situation of consistent decline without a recovery.

Step 4: Assess Logistic Growth

The logistic model, used for population that levels off at a carrying capacity, is not appropriate here since the population is decreasing, not stabilizing.

Step 5: Choose the Appropriate Model

An exponential decay model fits the situation as population decrease could initially be rapid following the industry's closure, then slow as the remaining population stabilizes.

Step 6: Determine Domain Restrictions

The domain of the function should be restricted to non-negative time values starting from when the industry closes. Additionally, since population cannot be negative, the model should only be used until the population approaches zero or reaches a stable state.

Key Concepts

Exponential Decay

Exponential decay describes a process in which a quantity decreases at a rate proportional to its current value. In layman's terms, this means that as the quantity shrinks, the rate at which it decreases becomes slower.

Here is how it applies to population:

• The population decreases rapidly right after the principal industry closes, leaving people jobless and prompting them to move elsewhere.
• As time passes, fewer people remain in the city, which means fewer people are leaving.
• This leads to a slowing down of the overall population decline.

Conversely, exponential growth functions show the opposite, where growth happens at a rate proportional to the current value. For population decline, exponential decay models are suitable as they accurately capture the declining rate post the initial rapid drop. To represent this mathematically, the exponential decay of a population can be expressed with the function: \[ P(t) = P_0 \cdot e^{-kt} \]where:

• \(P(t)\) is the population at time \(t\)
• \(P_0\) is the initial population when the industry closes
• \(k\) is the decay constant, a positive number
• \(t\) is the time elapsed since the industry closed

This equation tells us how to predict future population levels based on current and initial states.

Population Decline

Population decline refers to a reduction in a population's size over time. This can occur due to various factors, including economic downturns, increased emigration, or lower birth rates. When an area's main industry—like a city's principal employer—closes, it often leads to population decline.

Here’s how this unfolds:

• Job losses mean people have lesser means to sustain themselves, prompting many to move away in search of better opportunities.
• As people leave, the demand for housing, services, and local infrastructure decreases, causing further economic downturn.
• The remaining population might stabilize after an initial exodus but tends to be smaller than before the industry closure.

This pattern of population change can exhibit exponential decay, especially as the departures are rapid initially and then slow.

It's essential to model such declines accurately to plan for infrastructure, social services, and economic recovery effectively. Understanding these trends helps in creating support mechanisms for those affected.

Domain Restrictions

Domain restrictions are essential in mathematical modeling to define the boundaries and applicability of the model. When dealing with population decline due to an industry closure, these restrictions become particularly important.

In our model of exponential decay, imposing domain restrictions ensures our function remains realistic and meaningful:

• Time should be non-negative; it starts counting from when the industry closure occurs, therefore \(t \geq 0\).
• The population must also remain non-negative through the model, which prevents projecting nonsensical negative population numbers.
• The model primarily holds value only as long as it accurately represents the real-world scenario—in this case, until the city potentially reaches a population plateau or stabilizes.

These restrictions on the domain help keep our model scientifically sound and realistically applicable to the scenario being studied.