Problem 21 - 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. (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) 26.9 million tons/year; (c) 1241 million tons in 1975; (d) 2317 million tons; (e) Year 2022.

Step 1: Identify Variables and Data Points

We have two data points: \( (0, 1241) \) and \( (30, 2048) \). Here, \( t = 0 \) represents the year 1975, and \( t = 30 \) represents the year 2005.

Step 2: Calculate the Slope

The slope of the line (rate of change) is given by the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \). For our points, \( (0, 1241) \) and \( (30, 2048) \):\[ m = \frac{2048 - 1241}{30 - 0} = \frac{807}{30} \approx 26.9 \text{ million tons per year} \].

Step 3: Write the Linear Equation

Using the point-slope form \( y = mx + b \), where \( m = 26.9 \) and \( b \) is the vertical intercept (when \( t = 0 \) or year 1975):\[ P(t) = 26.9t + 1241 \].

Step 4: Interpret the Slope

The slope of 26.9 indicates that world grain production has been increasing by approximately 26.9 million tons each year since 1975.

Step 5: Interpret the Vertical Intercept

The vertical intercept of 1241 million tons indicates that the estimated world grain production in the base year 1975 was 1241 million tons.

Step 6: Predict Grain Production in 2015

2015 corresponds to \( t = 40 \) years since 1975. Substitute \( t = 40 \) into the equation:\[ P(40) = 26.9 \times 40 + 1241 = 2076 + 1241 = 2317 \text{ million tons} \].

Step 7: Determine When Production Reaches 2500 Million Tons

To find when production reaches 2500 million tons, set \( P(t) = 2500 \) and solve for \( t \):\[ 2500 = 26.9t + 1241 \ 1259 = 26.9t \ t \approx \frac{1259}{26.9} \approx 46.8 \].Round to the nearest whole number, production is predicted to reach 2500 million tons in year 1975 + 47 = 2022.

Key Concepts

Slope Interpretation

In the context of linear functions, the slope is a crucial concept. It represents the rate of change. In our exercise about grain production, the slope tells us how much grain production increases each year.
The formula to calculate the slope is \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]Using the given data points: (0, 1241) and (30, 2048), the slope was calculated as 26.9.
This implies:

• Each year, the world grain production increases by about 26.9 million tons.
• The slope reflects growth over time, which is constant in linear functions.

Understanding the slope helps in making predictions about future production, which is aligned with linear trends. This is why the slope is an essential part of our linear equation.

Vertical Intercept

The vertical intercept, also known as the y-intercept, is a foundational element in interpreting linear equations. It represents the output value when the input (in this case, the number of years since 1975) is zero.
In the grain production exercise, the vertical intercept is 1241, meaning:

• In 1975 (our starting point), the estimated grain production was 1241 million tons.
• This intercept provides a baseline for all future predictions.

The formula for a line is often written as \( y = mx + b \), where \( b \) is the vertical intercept. For the equation derived from our exercise, \( P(t) = 26.9t + 1241 \), the intercept helps affirm the grain production level at the beginning of the observation. By understanding it, one can better appreciate the growth trend from this starting point.

Grain Production Prediction

Predicting future grain production involves applying our linear function to specific years. This is precisely why we find the linear equation useful for projections.
In the exercise:

• To predict production in 2015, we set \( t = 40 \), as 2015 is 40 years after 1975.
• By substituting into the equation \( P(t) = 26.9t + 1241 \), we calculated \( P(40) = 2317 \) million tons.

Such predictions rely on our assumption that production rates remain constant. Additionally, the model forecasts when production will reach a significant milestone, like 2500 million tons. Solving \( 2500 = 26.9t + 1241 \) shows that this is expected in 2022. These calculations highlight the power of linear models in forecasting and decision-making.