Problem 44 - Calculus and its applications

Question

Farmers wishing to avoid the use of genetically modified (GMO) seeds are increasingly concerned about inadvertently growing GMO plants as a result of pollen drifting from nearby farms. Assuming that these farmers raise their own seeds, the fractional portion of their crop that remains free of GMO plants \(t\) years later can be approximated by \(P(t)=(0.98)^{t}\). a) Using this model, predict the fractional portion of the crop that will be GMO-free 10 yr after a neighboring farm begins to use GMO seeds. b) Find \(P^{\prime}(15)\). c) Interpret the meaning of \(P^{\prime}(15)\).

Step-by-Step Solution

Answer

a) 81.7% of the crop remains GMO-free after 10 years. b) \( P'(15) ≈ -0.0123 \). c) The GMO-free fraction decreases by about 1.23% per year at 15 years.

Step 1: Understand the Given Model

The problem provides a model for predicting the fractional portion of a crop that remains GMO-free over time. This model is given by the function \( P(t) = (0.98)^t \), where \( t \) is the number of years after a neighboring farm begins using GMO seeds. Our goal is to apply this model to solve the given problems.

Step 2: Predicting the Fractional Portion for Part a

To predict the fraction of the crop that remains GMO-free 10 years after GMO seeds begin nearby, substitute \( t = 10 \) into the model: \[P(10) = (0.98)^{10}\]. Upon calculation: \[P(10) ≈ 0.817\]. Thus, approximately 81.7% of the crop will remain GMO-free after 10 years.

Step 3: Finding the Derivative for Part b

To find the derivative \( P'(t) \), we start with \( P(t) = (0.98)^t \), which is an exponential function of the form \( a^t \). The derivative of \( a^t \) is \( a^t \, \ln(a) \). Here, \( a = 0.98 \), so: \[P'(t) = (0.98)^t \, \ln(0.98)\]. Next, substitute \( t = 15 \) into this derivative to find \( P'(15) \): \[P'(15) = (0.98)^{15} \, \ln(0.98)\]. Calculating this gives: \[P'(15) ≈ -0.0123\].

Step 4: Interpret the Meaning of the Derivative for Part c

The derivative \( P'(15) \) represents the rate of change of the GMO-free fraction of the crop at \( t = 15 \) years. The negative value \( P'(15) ≈ -0.0123 \) indicates that at 15 years, the portion of the crop that is GMO-free is decreasing by about 1.23% per year.

Key Concepts

Derivative Calculation

When dealing with exponential decay models like our function \( P(t) = (0.98)^t \), understanding the derivative is crucial. The derivative tells us how quickly the function's value is changing over time.
In general, the process of finding the derivative of an expression like \( a^t \) involves using the formula \( a^t \ln(a) \). Here, \( a = 0.98 \), which means:
\[ P'(t) = (0.98)^t \ln(0.98) \]
The derivative is a key concept in calculus that models the instantaneous rate of change. It's like asking, "How is the situation evolving right now?" in mathematical terms. By substituting \( t = 15 \) into the equation, we calculate:
\[ P'(15) = (0.98)^{15} \ln(0.98) \approx -0.0123 \]
This negative derivative value helps us understand what’s happening to our model at a very specific moment. It's like checking the pulse of the function at \( t = 15 \) to see its trend.

Model Interpretation

Models help us predict and understand real-world situations. The model provided, \( P(t) = (0.98)^t \), allows us to predict how much of the crop remains GMO-free as time passes.
In this model:

• The base \( 0.98 \) indicates a decrease. The model is telling us that each year, only 98% of what was GMO-free remains so from the previous year.
• As \( t \) increases, \( P(t) \) will get smaller because \( (0.98)^t \) tends towards zero.

Understanding models like these requires recognizing the patterns they show about change over time. Here, the model is a powerful tool for farmers, as it helps predict the long-term implications of GMO seed use nearby.

Rate of Change

The rate of change helps us understand how a situation is unfolding. In our exercise, we’re concerned with how quickly the fraction of GMO-free crops is changing after 15 years.
The rate of change, given by \( P'(15) \approx -0.0123 \), tells us that the fraction of the crop that remains GMO-free is decreasing.

• A negative rate of \( -0.0123 \) implies a downward trend, showing a decrease in the GMO-free portion each year.
• Numbers like \( -0.0123 \) represent percentages, meaning a 1.23% annual decrease in GMO-free crops.

Recognizing the rate of change helps farmers make informed decisions about managing their crops and seed selection strategies long term. Monitoring these changes can guide them in responding to the environmental shifts around them.