Problem 22 - Applied Calculus

Question

World grain production was 1241 million tons in 1975 and 2048 million tons in 2005, and has been increasing at an approximately constant rate. \({ }^{15}\) (a) Find a linear function for world grain production, \(P\), in million tons, as a function of \(t\), the number of years since 1975 . (b) Using units, interpret the slope in terms of grain production. (c) Using units, interpret the vertical intercept in terms of grain production. (d) According to the linear model, what is the predicted world grain production in 2015 ? (e) According to the linear model, when is grain production predicted to reach 2500 million tons?

Step-by-Step Solution

Answer

(a) \( P(t) = 26.9t + 1241 \); (b) Increase of 26.9 million tons/year; (c) Production of 1241 million tons in 1975; (d) 2317 million tons; (e) Approximately in 2022.

Step 1: Identify Variables and Data Points

We are given two data points: grain production was 1241 million tons in 1975 and 2048 million tons in 2005. We define a linear function for grain production, \( P(t) = mt + b \), where \( t \) is the number of years since 1975.

Step 2: Calculate the Slope

To find the slope \( m \), we use the formula for the slope of a line: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Substituting the known values, \( y_2 = 2048 \), \( y_1 = 1241 \), \( x_2 = 30 \) (since 2005 is 30 years after 1975), and \( x_1 = 0 \) (since 1975 is 0 years after 1975), we get: \[ m = \frac{2048 - 1241}{30 - 0} = \frac{807}{30} = 26.9 \] Thus, the slope \( m \) is 26.9 million tons per year.

Step 3: Determine the Vertical Intercept

The vertical intercept \( b \) is the grain production in 1975, which was 1241 million tons. Therefore, the linear function is: \[ P(t) = 26.9t + 1241 \]

Step 4: Interpret the Slope

The slope of 26.9 million tons per year means that the world grain production is increasing by approximately 26.9 million tons each year.

Step 5: Interpret the Vertical Intercept

The vertical intercept, 1241 million tons, represents the world grain production at the starting year of 1975.

Step 6: Predict Production in 2015

To find the grain production in 2015, we substitute \( t = 40 \) (since 2015 is 40 years after 1975) into the linear function: \[ P(40) = 26.9 \times 40 + 1241 = 1076 + 1241 = 2317 \] Thus, the predicted production is 2317 million tons in 2015.

Step 7: Find the Year Production Reaches 2500 Million Tons

Set the function equal to 2500 and solve for \( t \): \[ 26.9t + 1241 = 2500 \] \[ 26.9t = 2500 - 1241 \] \[ 26.9t = 1259 \] \[ t = \frac{1259}{26.9} \approx 46.8 \] Since \( t \) is 46.8 years after 1975, the production is predicted to reach 2500 million tons in approximately 2022.

Key Concepts

Slope Interpretation in Context

Understanding the slope in a linear function is crucial for interpreting the rate of change in a given context. In the scenario of world grain production, the slope equates to 26.9 million tons per year. This value illustrates how much the production grows annually.
This increase in grain production is consistent year over year, due to the linear nature of the model.

• What does this mean practically? A slope of 26.9 tells us that every single year, the global grain yield is expected to rise by 26.9 million tons compared to the previous year.

Such interpretive power allows us to quantify the growth rate effectively in real-world terms. Thus, observing this slope gives stakeholders insights into how sustainable or rapid agricultural advancements are. Highlighting the understanding of slope interpretation helps to predict future trends and make informed policy and investment decisions.

Vertical Intercept in Context

The vertical intercept in linear functions often tells us the initial condition for the model. For the world grain production model, the vertical intercept is at 1241 million tons.

• Why is it significant? This number represents the total grain production at the starting point of 1975.
• Real-world implication: If no other variables changed, such as farming practices or weather conditions, 1241 million tons would be the amount existing at the benchmark year of this model.

These details anchor the linear model to a specific starting point and help validate the function's effectiveness over time. Understanding the vertical intercept equips policymakers and analysts with baseline data against which growth metrics can be compared.

Predictions with Linear Models

One of the main uses of linear models in applied calculus is making predictions about future events based on current trends. Using our model of world grain production, predictions can be quickly calculated.
Plugging the desired year into the linear equation helps us estimate production levels. For example, when we plug in 40 years to find the production in 2015, we calculated a total of 2317 million tons.

• These predictions are vital for planners and policymakers, enabling better resource distribution decisions.
• Forecasts can also help anticipate future stresses or shortages in grain production, if the trend were to change drastically.

By understanding how to use linear models to project future outcomes, stakeholders can craft more resilient and proactive strategies.

Mathematical Modelling in Real-World Contexts

Mathematical modeling allows us to represent complex phenomena in a simplified, theoretical framework. The basis for our world grain production model uses a linear equation, which serves as a powerful tool for simplifying gradual trends.

• Core idea: The model captures the rate of increase as a straight line rather than handling each incremental change separately.
• Benefits for stakeholders: Models like this can support strategic planning, policy-making, and resource allocation decisions by providing clear expectations of future trends.
• Connection to real changes: A linear model works well as long as the changes stay steady. If unexpected factors cause rapid shifts, models might need modifications.

The advantage of mathematical modeling is that it transforms real-world challenges into manageable mathematical problems, allowing for structured problem-solving and strategic forecasting.