Problem 53 - 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 drop and rise of a lake's water level during and after a drought

Step-by-Step Solution

Answer

A logistic model is appropriate due to the phases of decline and rapid recovery with domain restrictions from the start of drought to stabilization.

Step 1: Analyze the Situation

To determine the appropriate model, we must consider the dynamics of the lake's water level. During a drought, the water level decreases gradually, and when the drought ends, the level might increase rapidly due to rainfall or snowmelt.

Step 2: Choose the Appropriate Model

The scenario suggests two phases: a gradual decrease and a potentially rapid increase in the water level. The decrease might be better modeled by a linear or logistic decline, while the subsequent rise due to environmental replenishment seems exponential. Thus, a logistic growth model is suitable as it can model both decrease and saturation points effectively.

Step 3: Domain Restrictions

The domain is restricted to the period during and after the drought. This includes the duration of the drought as well as a recovery period until the lake reaches a stable level again. Excluding negative values makes sense, as water levels cannot be negative.

Key Concepts

Logistic Growth

When trying to understand real-world phenomena, such as changes in a lake's water levels during and after a drought, we often turn to mathematical modeling. Logistic growth is a type of model that captures how populations or quantities adjust over time, considering factors like limited resources or carrying capacities. In the context of a lake during a drought, logistic growth might initially seem unconventional. However, it effectively describes situations where there is a gradual change followed by fluctuations around a stable level.
A logistic model incorporates phases of growth (like water level rise) and then plateaus as it approaches a carrying capacity. These phases are aptly suited for natural situations where external influences, such as rainfall replenishing the lake, eventually slow down as the environment stabilizes.
Logistic models are represented by the equation:

• \( P(t) = \frac{K}{1 + \frac{C - 0}{C} e^{-rt}} \)

Here, \( K \) is the carrying capacity, \( C \) is the initial quantity, and \( r \) is the growth rate. This formula shows how initial rapid growth ultimately slows as the level approaches a steady state.

Domain Restrictions

Just as in a mathematical function, applying a domain restriction means defining where this model is valid. In the case of a lake's water level, domain restrictions are necessary to ensure the model accurately reflects the real situation.
For logistic growth models, the domain should be limited to the period surrounding the drought and the subsequent recovery phase. This means the model should start at the onset of the drought and extend until the water level stabilizes post-drought.
It is also intuitive to exclude negative values from the domain—since negative water levels are not possible in reality. Therefore, setting these restrictions ensures the model remains relevant and meaningful, providing accurate predictions of the lake's behavior during the defined time.

Exponential Functions

Exponential functions describe processes that grow or decay at a constant relative rate. They are a natural fit for the rapid increase of a lake's water levels post-drought, driven by heavy rainfall or snowmelt.
Unlike logistic models that incorporate an upper limit, exponential functions assume continued growth or decline without such constraints. Here is a basic form of an exponential equation:

• \( N(t) = N_0 \, e^{kt} \)

In this equation, \( N(t) \) represents the quantity at time \( t \), \( N_0 \) is the initial amount, and \( k \) is the growth/decay rate.
In the context of the lake, an exponential model might overshoot as it does not account for a saturation point, which logistic models do. However, for a short time frame, exponential growth is efficient in depicting rapid changes once limiting factors are temporarily removed, like during the initial heavy rains following a drought.