Problem 58 - A Graphical Approach to College Algebra

Question

The following table shows the average Valentine's Day spending in dollars per consumer for various years. $$\begin{array}{|l|c|c|c|c|} \hline \text { Year } & 2010 & 2011 & 2012 & 2013 \\\ \hline \text { Spending ( } \$ \text { ) } & 103 & 116 & 126 & 131 \\\\\hline\end{array}$$ (a) Use exponential regression to approximate values for \(a\) and \(b\) so that \(f(x)=a+b \ln x\) models the data, where \(x=1\) corresponds to \(2010, x=2\) to 2011 and so on. (b) Use your function to estimate average spending in 2012 and compare to the value in the table.

Step-by-Step Solution

Answer

The model estimates lower spending for 2012 than the table shows.

Step 1: Set Up Variables and Equations

We need to model the data using the function \(f(x) = a + b \ln x\). First, we assign corresponding \(x\) values for each year: \(x = 1\) for 2010, \(x = 2\) for 2011, \(x = 3\) for 2012, and \(x = 4\) for 2013. Then, we express spending \(S\) in terms of this function: \(f(1) = 103\), \(f(2) = 116\), \(f(3) = 126\), and \(f(4) = 131\).

Step 2: Set Up the System of Equations

Using each point (\(x\), spending), we set up the following system: \[ \begin{align*} &103 = a + b \ln(1), \ &116 = a + b \ln(2), \ &126 = a + b \ln(3), \ &131 = a + b \ln(4). \ \end{align*} \] The first equation simplifies to \(103 = a\) since \(\ln(1) = 0\).

Step 3: Solve for Coefficients

From \(103 = a\), we have \(a = 103\). Substitute \(a\) into the subsequent equations:\[ \begin{align*} &116 = 103 + b \ln(2), \ &126 = 103 + b \ln(3), \ &131 = 103 + b \ln(4). \ \end{align*} \] Simplifying the second equation gives \(116 - 103 = b \ln(2)\), so \(b = \frac{13}{\ln(2)}\). Similarly, solve for \(b\) using the other equations and take the average value of \(b\) to obtain a more accurate model.

Step 4: Calculate the Average Value of b

Calculate \[\begin{align*} b_2 &= \frac{13}{\ln(2)}, \ b_3 &= \frac{23}{\ln(3)}, \ b_4 &= \frac{28}{\ln(4)}. \end{align*}\] Then compute\[ b = \frac{b_2 + b_3 + b_4}{3}. \] Insert the calculated values and compute \(b\).

Step 5: Use the Model to Estimate Spending for 2012

With \(a = 103\) and calculated \(b\), the function becomes \(f(x) = 103 + b \ln(x)\). For 2012 (\(x = 3\)), calculate \(f(3) = 103 + b \ln(3)\) and compare it to the observed value 126.

Key Concepts

Average Spending

Understanding the concept of average spending is fundamental to analyzing consumer habits over time. Average spending simply refers to the mean amount of money spent by each person within a specified time frame. In this case, average Valentine's Day spending per person was recorded over several years.
- 2010: $103 - 2011: $116 - 2012: $126 - 2013: $131
Analyzing these averages over time helps businesses and analysts understand trends and make predictions about future spending habits. By observing that spending has increased each year, we can hypothesize potential reasons, such as increased consumer enthusiasm towards Valentine's Day or improvements in economic conditions. This concept plays a significant role when developing mathematical models like exponential regression to forecast future trends.

Logarithmic Function

The logarithmic function is an important mathematical tool used to model relationships that grow at a decreasing rate. In this exercise, we use a specific form of function: \( f(x) = a + b \ln(x) \).
Here, \(f(x)\) represents the average spending, \(a\) is a constant, and \(b\) is the coefficient of the logarithmic term. The natural logarithm \(\ln(x)\) is used instead of a basic logarithm because it simplifies calculations when dealing with growth processes often found in business and science contexts.
- **Why Use a Logarithmic Function?**
This equation is particularly valuable because it provides a simple way to model data that increases at rates that slow down as time progresses. In this case, it helps us understand the stabilizing growth of Valentine's Day spending over years and predicts future spending patterns. By using logarithmic regression, we can get more stable outputs, which are pertinent when real-world data does not follow a straight linear path.

Data Modeling

Data modeling involves creating mathematical models to represent real-world data, enabling analysts to interpret and predict future trends effectively. In this exercise, exponential regression is used to model the recorded spending data over years to understand and predict future spending.
- **Exponential Regression**: Although log functions are emphasized here, the method resembles exponential regression because it suggests exponential relationships are nicely modeled with logarithms.
- **Steps in Data Modeling**: - First, translate years into numerical \(x\) values - Create system of equations relating \(x\) values to observed spending - Solve to find coefficients \(a\) and \(b\) for the model
Once the model is established, it is used to predict average spending for given years, as seen in estimating for 2012:\[ f(3) = 103 + b \ln(3) \]By comparing the estimate with actual spending, analysts adjust their models to improve prediction accuracy. Learning how to build and interpret these models is crucial for data-driven decision-making in multiple industries.