Problem 26 - Applied Calculus

Question

(a) Could the data on annual world soybean production \(^{54}\) in Table \(1.32\) correspond to a linear function or an exponential function? If so, which? (b) Find a formula for \(P\), world soybean production in millions of tons, as a function of time, \(t\), in years since 2000 . (c) What is the annual percent increase in soybean production? $$ \begin{array}{l} \text { Table 1.32 Soybean production, in millions of tons }\\\ \begin{array}{c|c|c|c|c|c|c} \hline \text { Year } & 2000 & 2001 & 2002 & 2003 & 2004 & 2005 \\ \hline \text { Production } & 161.0 & 170.3 & 180.2 & 190.7 & 201.8 & 213.5 \\\ \hline \end{array} \end{array} $$

Step-by-Step Solution

Answer

Exponential growth; \(P(t) = 161 \times 1.058^t\); 5.8% annual increase.

Step 1: Examine Data for Linear or Exponential Growth

To determine if the soybean production data could correspond to a linear or exponential function, first look at the differences between consecutive yearly productions. These are: 9.3, 9.9, 10.5, 11.1, and 11.7. Since these differences are increasing, the data does not fit a linear function well. Next, check for approximate constant ratio between consecutive productions to assess exponential growth. The ratios (previous year's production) are approximately: 1.057, 1.058, 1.058, 1.058, and 1.058. These ratios suggest an exponential growth pattern.

Step 2: Determine the Type of Function

Since the differences between consecutive years in production are increasing and the ratio remains relatively constant at around 1.058, the data fits an exponential function better than a linear function.

Step 3: Formulate the Exponential Equation

For an exponential function, the formula is typically of the form \[ P(t) = P_0 imes (1 + r)^t \] where \( P_0 \) is the initial production, \( r \) is the growth rate, and \( t \) is the number of years since 2000. Here, \( P_0 = 161 \) and average growth rate, \( r \), which is the approximate value of 0.058 from earlier calculations.

Step 4: Write the Formula for Soybean Production

Using the exponential formula, the world soybean production can be expressed as: \[ P(t) = 161 imes (1.058)^t \]This equation models the production in millions of tons as a function of time \(t\), where \(t\) is the number of years since 2000.

Step 5: Calculating the Annual Percent Increase

The growth factor per year is about 1.058, which implies an annual percent increase. To find the percent increase, subtract 1 from the growth factor and multiply by 100:\[ (1.058 - 1) imes 100 = 5.8\% \]The annual percent increase in soybean production is approximately 5.8%.

Key Concepts

Linear vs Exponential Growth

When analyzing data, it's crucial to understand the difference between linear and exponential growthpatterns. Linear growth means that With each unit of time, we increase by a fixedamount. The equation of a linear functionis typically \( y = mx + b \),where \( m \) is the constant rate of change.On the other hand, exponential growth means the rate at which the data grows is proportional to its size. The equation for an exponential function is \( y = a \times (1 + r)^t \),where \( r \) is the growth rate. To determine if the soybean data corresponds tolinear or exponential growth,we need to look at the yearly differences and ratios:

• For linear growth, the differences between consecutive years should be constant.
• For exponential growth, the ratio of each year’s production over the previous yearshould be constant.

In the soybean production data,the differences aren't constant, but the ratios are.This patterned ratio of approximately 1.058 signals exponential growth.

Soybean Production

Soybean production is a critical aspect of global agriculture. The data from the early 2000s shows significant growth in production. Understanding this growth helps in planning for future food requirements and sustainable farming practices. Soybeans are widely used for:

• Animal feed, especially for livestock.
• Human consumption, including products like tofu and soy milk.
• Industrial uses such as biofuels and cosmetics.

Given their importance, it’s crucial to model soybean production accurately. In the exercise, the production from 2000 to 2005 increases from 161 to 213.5 million tons. This substantial increase necessitates an understanding of how production is growing, whether it's linearly or exponentially, to make informed decisions in agriculture policy and trade.

Exponential Growth Rate

Exponential growth, commonly observed in areas likefinance and population studies,requires careful determination of the growth rate.For soybeans, we see an exponential pattern,meaning each year, production increases by a certain percentage.The formula used is usually \( P(t) = P_0 \times (1 + r)^t \):

• \( P_0 \) is the starting amount (161 million tons in this case).
• \( r \) is the exponential growth rate;a constant factor by which the quantity multiplies.
• \( t \) represents time.

By observing the ratio of consecutive production figures,the growth rate \( r \) was calculated as approximately 0.058or 5.8% when expressed as a percentage,showcasing consistent annual growth in production.

Annual Percent Increase

The concept of an annual percent increase isessential for understanding trends and making predictions.In this context, it refers to how much soybean production grows each year,expressed as a percentage. To calculate it, we analyze the growth factor,which is around 1.058 in our soybean data.Subtract 1 to understand the actual increase:\[ 1.058 - 1 = 0.058 \]Multiply by 100 to get the percent: \[ 0.058 \times 100 = 5.8\% \] Hence, the world soybean production is increasing at an annual rate of 5.8%.This insight is vital for stakeholders, including farmers, policymakers, and economists,as it allows them to plan for future demand and adjust strategies accordingly.