Problem 106 - A Graphical Approach to Precalculus with Limits

Question

The table below shows the acreage, in millions, of the total of com and soybeans harvested annually in the United States. In the table, \(x\) represents the year and \(f\) computes the total number of acres for these two crops. The function \(g\) computes the number of acres for corn only. $$\begin{array}{c|c|c|c|c}\hline x & 2013 & 2014 & 2015 & 2016 \\\\\hline f(x) & 175.1 & 176.4 & 174.0 & 177.8 \\\\\hline g(x) & 97.4 & 91.6 & 88.9 & 94.1\end{array}$$ (a) Make a table for a function \(h\) that is defined by the equation \(h(x)=f(x)-g(x)\) \(\Rightarrow\) (b) Interpret what \(h\) computes.

Step-by-Step Solution

Answer

(a) The table for \(h(x)\) is: 2013: 77.7, 2014: 84.8, 2015: 85.1, 2016: 83.7. (b) \(h\) computes the number of acres of soybeans harvested annually.

Step 1: Understand the Problem

We are given a table with annual data for the total acres of corn and soybeans (function \(f(x)\)) and only corn (function \(g(x)\)) from 2013 to 2016. Our task is to compute a new function \(h(x)\) defined as \(h(x) = f(x) - g(x)\) for each year in the table.

Step 2: Calculate \(h(x)\) for Each Year

To find \(h(x)\), subtract \(g(x)\) from \(f(x)\) for each year:- For 2013: \(h(2013) = f(2013) - g(2013) = 175.1 - 97.4 = 77.7\)- For 2014: \(h(2014) = f(2014) - g(2014) = 176.4 - 91.6 = 84.8\)- For 2015: \(h(2015) = f(2015) - g(2015) = 174.0 - 88.9 = 85.1\)- For 2016: \(h(2016) = f(2016) - g(2016) = 177.8 - 94.1 = 83.7\)

Step 3: Create the Table for \(h(x)\)

Using the calculations from Step 2, construct a table for \(h(x)\):\[\begin{array}{c|c|c|c|c}\hline x & 2013 & 2014 & 2015 & 2016 \\hline h(x) & 77.7 & 84.8 & 85.1 & 83.7 \\end{array}\]

Step 4: Interpret \(h(x)\)

The function \(h(x)\) calculates the number of acres of soybeans harvested annually in the United States. This is because \(f(x)\) is the total acres of corn and soybeans combined, and \(g(x)\) is only the acres of corn, so \(h(x)\) isolates the acres for soybeans.

Key Concepts

Function Analysis

Analyzing functions involves understanding what each function represents and how they interact with each other. In the context of this exercise, you are given two functions:

• Function \( f(x) \): This function represents the total number of acres of corn and soybeans harvested each year.
• Function \( g(x) \): This function represents the number of acres of corn harvested each year.

These functions help you isolate data and understand the dynamics between the two types of crops.
To create the function \( h(x) = f(x) - g(x) \), you are subtracting the value of \( g(x) \) from \( f(x) \). Doing so, reveals specific data you need: the harvested acres of soybeans.
Using function analysis allows you to understand a broader dataset by focusing on specific elements like soybeans.

Table Data Interpretation

Interpreting table data is a crucial skill. It allows you to extract useful information from a structured set of numbers. In this exercise, the table shows you how different types of acres change over the years.
To interpret this effectively, you must identify each element:

• \( x \) is the year, indicating the time frame of the data.
• \( f(x) \) corresponds to the total acres of both corn and soybeans combined.
• \( g(x) \): relates specifically to the acres of corn.

By creating \( h(x) \) and looking at the table, you see fluctuations in soybean acres over the four years. Recognizing these patterns can help in planning and decision-making processes related to crop production.

Arithmetic Operations

Arithmetic operations are fundamental tools you use to manipulate numbers and understand relationships in data. In this problem, subtraction plays the critical role. It allows you to separate mixed data into its components.
Consider the calculation: \( h(x) = f(x) - g(x) \) happens individually for each year:

• For 2013: \( h(2013) = 175.1 - 97.4 = 77.7 \)
• For 2014: \( h(2014) = 176.4 - 91.6 = 84.8 \)
• For 2015: \( h(2015) = 174.0 - 88.9 = 85.1 \)
• For 2016: \( h(2016) = 177.8 - 94.1 = 83.7 \)

Each calculation directly impacts your understanding of the trend. You see how arithmetic operations clarify and refine large sets of data into manageable and insightful pieces.