Problem 110 - College Algebra and Calculus: An Applied Approach

Question

The average monthly sales \(y\) (in billions of dollars) in retail trade in the United States from 1996 to 2005 can be approximated by the model \(y=-22+117 \ln t, \quad 6 \leq t \leq 15\) where \(t\) represents the year, with \(t=6\) corresponding to 1996\. (Source: U.S. Council of Economic Advisors) (a) Use a graphing utility to graph the model. (b) Use a graphing utility to estimate the year in which the average monthly sales first exceeded \(\$ 270\) billion. (c) Verify your answer to part (b) algebraically.

Step-by-Step Solution

Answer

Based on the graph and the solution of the logarithmic equation, the average monthly sales first exceeded $270 billion in the year corresponding to \(t = e^{292/117}\). This verifies the graphical estimate.

Step 1: Graph the Model

Use a graphing utility to plot the given model \(y=-22+117 \ln t\). This should produce a graph of a logarithmic function.

Step 2: Estimate the Year

Scan the graph for the point at which \(y\) (monthly sales in billions of dollars) first surpasses 270. This will give an estimate for the value of \(t\) (the year).

Step 3: Algebraic Verification

To verify your answer algebraically, solve the equation \(-22 + 117 \ln t = 270\) for \(t\). This involves isolating \(t\) on one side of the equation. First add 22 to both sides to get \(117 \ln t = 292\). Then divide both sides by 117 to get \(\ln t = 292/117\). Finally, apply the exponential function to both sides to solve for \(t\), which yields \(t = e^{292/117}\). The value of \(t\) calculated here should match the year estimated from the graph.

Key Concepts

Graphing Utilities

Graphing utilities are essential tools for visualizing mathematical models, especially those involving logarithmic functions. When you are given a model like \(y = -22 + 117 \ln t\), a graphing utility can help you plot the function to understand its behavior. The graph will show a curve that represents the logarithmic relationship between the year \(t\) and the average monthly sales \(y\).With a graphing utility, you can:

• Input the equation to generate a graph.
• Adjust the viewing window to focus on the interval of interest, in this case, \(6 \leq t \leq 15\).
• Pinpoint specific values on the graph, such as where sales exceed certain levels.

Remember to ensure your units and scales are set correctly for accurate visualization. This aids in accurately estimating values like the year when monthly sales first surpass \$270 billion.

Retail Sales Model

The retail sales model given by the equation \(y = -22 + 117 \ln t\) provides a mathematical framework for analyzing the trend in retail sales over a specific period. This model uses the natural logarithm function, which is effective for modeling growth that increases rapidly at first and then levels off over time.In the equation:

• \(y\) represents the average monthly sales in billions of dollars.
• The term \(-22\) could be an adjustment factor needed for the model to fit the actual data.
• \(117 \ln t\) indicates the impact of the time \(t\) in years on the sales, with logarithm \(\ln t\) capturing the growth trend.

This type of model is particularly useful for economic analyses, allowing predictions about future sales based on past trends. It can help businesses and policymakers make informed decisions by understanding how sales grow over time.

Algebraic Verification

To ensure the estimates from the graph are accurate, you can use algebraic verification. This involves solving the model equation to directly find the year when the sales exceeded a certain amount, like \$270 billion.Here's how you do it algebraically:

• Start with the equation: \(-22 + 117 \ln t = 270\).
• Rearrange it to isolate the logarithmic part: add 22 to both sides to get \(117 \ln t = 292\).
• Divide both sides by 117 to solve for \(\ln t\): it becomes \(\ln t = 292/117\).
• Use the exponential function to get \(t\): \(t = e^{292/117}\).

Calculating \(t\) gives you the numerical year, which should align with the estimate from the graph. This verification step ensures the model's predictions are reliable using a combination of numerical and visual methods.