Problem 10 - Applied Calculus

Question

The Tojolobal Mayan Indian community in Southern Mexico has available a fixed amount of land. \({ }^{16}\) The proportion, \(P\), of land in use for farming \(t\) years after 1935 is modeled with the logistic function $$ P=\frac{1}{1+3 e^{-0.0275 t}} . $$ (a) What proportion of the land was in use for farming in \(1935 ?\) (b) What is the long-run prediction of this model? (c) When was half the land in use for farming? (d) When is the proportion of land used for farming increasing most rapidly?

Step-by-Step Solution

Answer

(a) 25%; (b) 100%; (c) Around 1975; (d) Also around 1975.

Step 1: Find Proportion in 1935

To find the proportion of land in use for farming in 1935, we need to substitute \( t = 0 \) into the logistic function, since 1935 is the base year. The logistic function is given by:\[P = \frac{1}{1 + 3e^{-0.0275t}} \]Substitute \( t = 0 \):\[P = \frac{1}{1 + 3e^{-0.0275 \times 0}} = \frac{1}{1 + 3 \times 1} = \frac{1}{4}\]So, 0.25 or 25% of the land was in use for farming in 1935.

Step 2: Determine Long-Run Prediction

The long-run prediction of the model corresponds to the limit of \( P \) as \( t \) approaches infinity. As \( t \to \infty \), the term \( e^{-0.0275t} \) approaches zero. The function becomes:\[P = \frac{1}{1 + 3 imes 0} = 1\]Thus, in the long run, 100% of the land is predicted to be used for farming.

Step 3: Find when Half the Land was in Use

To find when half the land was in use, set \( P = \frac{1}{2} \) and solve for \( t \):\[\frac{1}{2} = \frac{1}{1 + 3e^{-0.0275t}}\]Cross-multiplying gives:\[1 + 3e^{-0.0275t} = 2\]Then:\[3e^{-0.0275t} = 1\]Solve for \( e^{-0.0275t} \):\[e^{-0.0275t} = \frac{1}{3}\]Taking the natural logarithm of both sides:\[-0.0275t = \ln\left(\frac{1}{3}\right)\]Solving for \( t \):\[t = \frac{-\ln\left(\frac{1}{3}\right)}{0.0275} \approx 39.85\]This implies half the land was in use around 1975 (since 1935 + 40 = 1975).

Step 4: Find When Proportion of Land is Increasing Most Rapidly

The proportion of land used for farming increases most rapidly at the inflection point of the logistic function. For a logistic growth model, the inflection point occurs when \( P = \frac{K}{2} \) where \( K = 1 \). Therefore, \( P = \frac{1}{2} \).As calculated in Step 3, this occurs around \( t = 39.85 \) years after 1935, so it happens around 1975.

Key Concepts

Proportion of Land Use

In the logistic model given, the proportion of land use refers to the fraction of the total land that is being utilized for farming over time. The model is expressed as a logistic function:\[ P = \frac{1}{1 + 3 e^{-0.0275t}} \]This function helps us understand how land use changes from a base year, which here is 1935. The logistic function is useful because:

• It starts from a lower initial proportion, reflecting limited use initially.
• Gradually increases as time progresses, showing more land being used.
• Eventually approaches a saturation level where most or all land is in use.

In this specific model, the starting point is found by setting \( t = 0 \), which corresponds to the year 1935. Calculations show that initially, 25% of the land was used for farming in 1935. This gradual increase in the proportion of land use illustrates a typical growth pattern controlled by the logistic function, reflecting limitations in resource expansion.

Long-Run Prediction

Long-run prediction, in the context of logistic growth, refers to the state that the system aims to reach as time continues indefinitely. This is also known as the carrying capacity of the system. In the logistic function model here, we use:\[ P = \frac{1}{1 + 3 e^{-0.0275t}} \]As \( t \) approaches infinity, the exponential term \( e^{-0.0275t} \) becomes negligible. The function simplifies to:\[ P = \frac{1}{1 + 0} = 1 \]This tells us that 100% of the land is predicted to eventually be used for farming. This carries significant implications:

• It establishes that under the given conditions, no land would be unused in the long term.
• Factors such as changes in agricultural technology or policies could impact reaching this prediction.
• The logistic function's ceiling ensures that predictions align with realistic boundaries of resource limitation.

A key takeaway is understanding that logistic curves often predict saturation points as part of many growth processes, making them essential in long-run forecasting.

Inflection Point

The inflection point in a logistic growth model delineates where the rate of increase changes from accelerating to decelerating. In simple terms, it's the peak moment of growth pace. This point can be critical for decision-making and resource optimization.In our function:\[ P = \frac{1}{1 + 3 e^{-0.0275t}} \]The inflection point occurs when \( P \) is halfway between its starting point and the predicted maximum. Since the maximum \( K = 1 \), the inflection happens when \( P = \frac{1}{2} \).Calculations show this inflection point occurs around the year 1975, nearly 40 years from 1935 (years = 1935 + 39.85). At this juncture, we might see:

• A maximal increase in farming activities.
• Possibly more emphasis on infrastructure or resource allocation.
• Strategic movements to handle impending full utilization post this phase.

Recognizing an inflection point in logistic growth provides insight into the timeline of changes, helping anticipate peaks and prepare for future stabilization.

Logarithmic Function

Logarithmic functions are the inverse of exponential functions and play a crucial role in solving exponential equations. In the context of logistic growth models, they help determine specific time moments where certain conditions, like the inflection point, occur.In solving for when the land is half-used, the equation set was:\[ \frac{1}{2} = \frac{1}{1 + 3e^{-0.0275t}} \]This results in solving:\[ e^{-0.0275t} = \frac{1}{3} \]Taking the natural logarithm of both sides is vital here:\[ -0.0275t = \ln\left(\frac{1}{3}\right) \]From which we solve for \( t \):\[ t = \frac{-\ln\left(\frac{1}{3}\right)}{0.0275} \approx 39.85 \]Using logarithms helps contextualize and compute when particular growth points occur:

• Provides a mathematical means to unravel time-dependent behaviors in growth models.
• Aids in shifting exponential growth perspectives into more manageable calculations.
• Essential for interpreting logistic models beyond basic intuition.

Thus, grasping logarithmic functions assists in unlocking deeper understanding of growth phenomena and predictions within logistic frameworks.