Question

The cumulative sales \(S\) (in thousands of units) of a new product after it has been on the market for \(t\) years are modeled by \(S=C e^{k / t}\) During the first year, 5000 units were sold. The saturation point for the market is 30,000 units. That is, the limit of \(S\) as \(t \rightarrow \infty\) is 30,000 . (a) Solve for \(C\) and \(k\) in the model. (b) How many units will be sold after 5 years? (c) Use a graphing utility to graph the sales function.

Short answer

The constants for the model are \(C=30000\) and \(k=ln(1/6)\). After 5 years, approximately \(S=30000e^{ ln(1/6)/5}\) units will be sold.

Step-by-step solution

Understanding the Model

This model shows that the cumulative sales (S) of the product are related to the time (t) it has been on the market. The limit of the model being the saturation point. Hence, S approaches 30,000 as t approaches infinity.

Solving for C and k

From the problem we know that 5000 units were sold in the first year. Hence when \(t=1\), \(S=5000\). We substitute these into the model :\(5000=Ce^{k}\). The limit as \(t \rightarrow \infty\) is 30000. Substitute into model and solve for C. \(30000=Ce^{0}\), that implies \(C=30000\). Hence, subsituting C into equation \(5000=30000e^{k}\), simplifying the equation, we get \(k= ln(1/6)\)

Units sold after 5 years

Now that we have C and k, we can know how many units will be sold after 5 years, substituting \(t=5\), \(C=30000\) and \(k=ln(1/6)\) into the model, we get \(S=30000e^{ ln(1/6)/5}\)

Visualizing the sales function

Although we can't draw the graph here, it's important to understand it will help visualize the growth of sales over time based on the model function. The function modeled in the graph will start at S=5000 at year 1 and approach S=30000 as time goes to infinity

Key concepts

Cumulative Sales

In the context of product sales, the term "cumulative sales" refers to the total number of units sold over a specific period. It accumulates over time as more units are sold. Understanding this is crucial for businesses as it helps in evaluating market performance and devising strategies.

Mathematically, in the given model, cumulative sales are represented by \( S = C e^{k/t} \), where:

• \( S \) stands for cumulative sales in thousands of units.
• \( C \) is a constant representing an initial scaling factor.
• \( e \) is the base of natural logarithms, approximately equal to 2.71828.
• \( k \) is a parameter that modifies how sales grow over time.
• \( t \) is the time in years.

In the exercise, it is given that 5000 units were sold in the first year, providing a point of reference to find \( C \) and \( k \). This equation helps in predicting sales at various points in the future based on historical data.

Saturation Point

The saturation point in sales refers to the maximum capacity of the market for a particular product. At this stage, sales plateau as the market can no longer accommodate an increase due to limitations such as full market penetration.

In the provided model, the saturation point is modeled as \( S \to 30000 \) units as \( t \to \infty \). This implies that no matter how long the product is available, the sales will not exceed 30,000 units.

Understanding the saturation point is vital for:

• Strategizing product launches and marketing efforts.
• Forecasting future sales and potential revenue.
• Deciding when to introduce new products to maintain market interest.

This concept also aids companies in reallocating resources efficiently once the market limit is nearly reached.

Graphing Sales Function

Graphing the sales function provides a visual representation of the cumulative sales over time, helping stakeholders understand trends and make informed decisions.

For the given model \( S = 30000 e^{\ln(1/6)/t} \) , graphing can demonstrate:

• The initial steep increase in sales as the product penetrates the market.
• The gradual flattening as the saturation point is approached.

Using a graphing utility, you can plot \( S \) against \( t \) to observe these dynamics clearly. The graph will begin at approximately 5000 units (at \( t=1 \)) and asymptotically approach 30,000 units over time.

This visualization aids in:

• Communicating the potential growth phases of the product to stakeholders.
• Identifying necessary actions if sales decline or stagnate earlier than expected.
• Spotting opportunities for marketing interventions to speed up sales.

Ultimately, graphing sales functions helps businesses visualize the impact of their sales strategies and adjust them to maintain healthy growth.