Problem 25 - College Algebra with Modeling and Visualization

Question

Between 1950 and 1980 the use of chemical fertilizers increased. The table lists worldwide average usage \(y\) in kilograms per hectare of cropland, \(x\) years after \(1950 .\) (Note: 1 hectare \(\approx 2.47\) acres.) $$\begin{array}{ccccc}x & 0 & 13 & 22 & 29 \\ \hline y & 12.4 & 27.9 & 54.3 & 77.1\end{array}$$ (a) Graph the data. Are the data linear? (b) Find a function \(f\) that models the data. (c) Predict fertilizer usage in \(1989 .\) The actual value was 98.7 kilograms per hectare. What does this indicate about usage of fertilizer during the 1980 s?

Step-by-Step Solution

Answer

Yes, data is linear; model: \(f(x) = 2.231x + 12.4\); 1989 prediction: 99.4 kg/hectare.

Step 1: Create a Scatter Plot

Plot the given data points on a graph to visualize the relationship between the years since 1950 (x-values) and the corresponding usage of chemical fertilizers in kilograms per hectare (y-values). This will help determine if the data is linear.

Step 2: Determine Data Linearity

Examine the scatter plot to assess if the data points fall approximately in a straight line. This can indicate if the data is linear.

Step 3: Choose a Model

Since data suggests a linear relationship, propose a linear function of the form \(f(x) = ax + b\) to model the data. We can use linear regression or select two points to determine the slope \(a\) and y-intercept \(b\).

Step 4: Compute the Slope

Select two data points to calculate the slope \(a\). For example, using points \((0, 12.4)\) and \((29, 77.1)\), compute the slope as follows: \[ a = \frac{77.1 - 12.4}{29 - 0} = \frac{64.7}{29} \approx 2.231 \]

Step 5: Calculate Y-Intercept

Using the slope \(a\) and one data point, solve for the y-intercept \(b\). Using \((0, 12.4)\): \[ f(0) = 2.231 \cdot 0 + b = 12.4 \Rightarrow b = 12.4 \]

Step 6: Write the Linear Function

With the slope \(a\approx 2.231\) and \(b = 12.4\), the linear function is: \[ f(x) = 2.231x + 12.4 \]

Step 7: Predict for 1989

Substitute \(x = 39\) (since 1989 is 39 years after 1950) into the function to predict fertilizer usage: \[ f(39) = 2.231 \cdot 39 + 12.4 \approx 99.4 \text{ kg/hectare} \]

Step 8: Interpret the Prediction

Compare the predicted value of 99.4 kg/hectare with the actual value of 98.7 kg/hectare. This close prediction indicates the linear trend continued into the early 1980s, suggesting potential stabilization in growth rate during the 1980s.

Key Concepts

Scatter Plot

A scatter plot is a powerful visual tool used to observe relationships between two quantitative variables. Think of it as a simple graph where each point represents a specific value of a dataset on the x and y coordinates. In our case, the x-values are years since 1950, and the y-values are the usage of fertilizers.
Creating a scatter plot involves plotting each pair of values (x, y) on a graph. This method helps to quickly determine if a relationship, such as linearity, exists between variables. If the points form a pattern resembling a straight line, it often indicates a linear relationship.
Knowing the nature of the relationship helps decide the mathematical model to use. By looking at our data from 1950 to 1980, when plotted, the points suggest a linear pattern, allowing us to proceed with calculating a linear function to model the data effectively.

Slope Calculation

Once the scatter plot suggests linearity, the next step is to calculate the slope of the line that best fits the data. The slope is a measure of how steep this line is and indicates how much the y-values increase (or decrease) as the x-values increase.
To calculate the slope, select two data points from the graph. For example, we can use the points

• (0, 12.4), which represents fertilizer usage in 1950,
• and (29, 77.1), which corresponds to 1979 usage.

Then, use the formula: \[ a = \frac{y_2 - y_1}{x_2 - x_1} \] Substituting the values gives: \[ a = \frac{77.1 - 12.4}{29 - 0} = \frac{64.7}{29} \approx 2.231 \]This slope of approximately 2.231 indicates that for each year increase in time since 1950, fertilizer usage increased by about 2.231 kilograms per hectare. Understanding this change is crucial for predicting future trends and making informed decisions about resource management.

Linear Function

After finding the slope, the next step is to construct the linear function that models the data. A linear function has the general form: \[ f(x) = ax + b \] where \( a \) is the slope, and \( b \) is the y-intercept. This equation represents how y changes with respect to x.
With our previously calculated slope, \( a \approx 2.231 \), and knowing a point

(0, 12.4)

, we use it to find the y-intercept \(b\) by substituting into the linear function: \[ f(0) = 2.231 \cdot 0 + b = 12.4 \]Thus, \( b = 12.4 \). So, the linear function becomes: \[ f(x) = 2.231x + 12.4 \]This function effectively models the fertilizer usage over time from 1950 to 1980. You can use this to predict future values by substituting different x-values. For example, predicting fertilizer usage for 1989, which is 39 years after 1950, we find: \[ f(39) = 2.231 \cdot 39 + 12.4 \approx 99.4 \text{ kg/hectare} \]This prediction confirms the data trend and helps guide expectations about changes in fertilizer usage over time. It's useful for both historical analysis and forecasting.