Question

The retail sales $S$ (in billions of dollars per year) of e-commerce companies in the United States from 1999 through 2004 are shown in the table. $$\begin{array}{|lllllll|}\hline t& 9& 10& 11& 12& 13& 14\\ S& 14.5& 27.8& 34.5& 45.0& 56.6& 70.9\\ \hline\end{array}$$ The data can be modeled by $S=-254.9+121.95\ln t$, where $t=9$ corresponds to 1999.

Short answer

To solve this problem, simply replace the given values into the logarithmic model and make calculations. For each year, the predicted sales model values should be pretty close to the actual sales values given in the table. Likewise, you can predict future sales by substituting the future years into the model.

Step-by-step solution

Understand the model

The model given, $S=-254.9+121.95\ln t$, is a logarithmic function, where $S$ represents the sales and $t$ represents the time in years. According to the problem, $t=9$ represents the year 1999.

Review the table

The table provides historical data for years (t’s values) ranging from 9 to 14 (1999 to 2004) and corresponding sales (S’s values) in billions. These values will help us to verify the adequacy of the model and can also be used to predict future sales.

Substitute the values

To verify the model, substitute the given values of $t$ from the table into the model and calculate the corresponding $S$ values. For instance, for $t=9$, we get $S=-254.9+121.95\ln 9$. Do this for all the given values of $t$ from the table and compare the calculated $S$ values with the given ones in the table for correctness.

Predict future sales

To predict future sales (e.g. for the year 2005, which is $t=15$), substitute $t=15$ into the model to find the corresponding $S$ value.

Key concepts

Algebraic Modeling with Logarithmic Functions

Algebraic modeling is an invaluable tool for illustrating real-world phenomena using equations. Specifically, logarithmic functions are particularly useful in modeling scenarios where growth decelerates over time. For example, in commerce, sales figures might initially grow rapidly but tend to stabilize as the market becomes saturated.

In the context of the exercise, the model provided, $S=-254.9+121.95\ln t$, uses a logarithmic function to approximate the sales of e-commerce companies over time. The algebraic model consists of two parts: the constant term $-254.9$ and the logarithmic term $121.95\ln t$. The constant term represents the sales when $t=0$, although this is not directly interpretable within the model's context, as the starting point is $t=9$. The coefficient $121.95$ scales the logarithmic term, which adjusts for the rate that the natural logarithm of time impacts the sales.

To ensure the model's reliability given new data, recalibration might be necessary if substantial discrepancies are observed when comparing actual sales figures against the model's predictions. Command of algebraic modeling enables students to understand the structure of such models and assess their real-world applications accurately.

Interpreting Logarithmic Models

Deciphering logarithmic models involves recognizing the implications of the logarithmic relationship between variables. The natural logarithm (ln), found in the given model, indicates a type of growth pattern that is common in business contexts, like market saturation or diminishing returns in sales and revenue over time.

From a commerce standpoint, interpreting the logarithmic model involves relating the logarithmic value of time, $\ln t$, to sales figures $S$. For each unit increase in $\ln t$, sales increase by the coefficient of the logarithmic term, which is $121.95$ in this case. This can be interpreted as the 'elasticity' of sales with respect to time – a concept widely used in economics referencing how sensitive a variable is to changes in another variable. It is important to remember the logarithm of a number is smaller for smaller values and increases at a decreasing rate for larger values of that number.

The model implies that each subsequent year does not add a constant amount to sales; instead, each year contributes less to the growth than the previous year, exemplifying the very nature of logarithmic growth patterns. However, while the logarithmic model provides insight, it should not be taken as an absolute predictor, especially if external factors likely to affect the market change.

Exponential and Logarithmic Functions in Commerce

Exponential and logarithmic functions are two sides of the same coin, with their core difference being how they quantify growth. While exponential functions model scenarios with constant percentage growth rates over equal time intervals, logarithmic functions often model the inverse scenario where growth rates decline over time.

In the commercial setting, understanding these functions is crucial since many economic behaviors exhibit exponential or logarithmic trends, whether concerning sales, interest rates, or market penetration. For instance, a savings account with a fixed interest rate compounds interest exponentially, whereas a maturing market's sales may increase in accordance with a logarithmic function as market penetration approaches its limit.

The given model $S=-254.9+121.95\ln t$ employs a logarithm to express sales over time, contrasting the more aggressive growth predicted by an exponential function. This model acknowledges that e-commerce sales won't indefinitely grow at the same rate and reflects the complexities of predicting economic and business activity using algebraic approaches.

It is key to educate students not only in the mechanics of using these functions but also in the interpretation of their parameters and implications. By doing so, it ensures that they can leverage these mathematical tools to forecast and analyze patterns in various business and economic contexts effectively.