Problem 3 - Applied Calculus

Question

The following table shows the total sales, in thousands, since a new game was brought to market. (a) Plot this data and mark on your plot the point of diminishing returns. (b) Predict total possible sales of this game, using the point of diminishing returns. $$\begin{array}{c|c|c|c|c|c|c|c|c}\hline \text { Month } & 0 & 2 & 4 & 6 & 8 & 10 & 12 & 14 \\\\\hline \text { Sales } & 0 & 2.3 & 5.5 & 9.6 & 18.2 & 31.8 & 42.0 & 50.8 \\\\\hline\end{array}$$

Step-by-Step Solution

Answer

The point of diminishing returns occurs around month 10. Estimated maximum sales are between 55 to 60 (in thousands).

Step 1: Create the Data Plot

First, create a scatter plot using the given data. Plot "Month" on the x-axis and "Sales" (in thousands) on the y-axis. Mark each data point to visualize how sales change over time.

Step 2: Analyze the Sales Growth Trend

Look at the plotted data to identify the growth trend. Notice how sales increase rapidly initially and then start to increase at a slower rate as time progresses.

Step 3: Identify the Point of Diminishing Returns

The point of diminishing returns is where the rate of increase starts to slow significantly. In the plotted graph, this may be observed around month 10, where sales growth begins to taper off.

Step 4: Extend the Trend to Predict Maximum Sales

Using the trend identified in the graph, predict the potential total sales. Assume sales will continue to grow slower until reaching a practical maximum. This point is generally a horizontal asymptote, representing the estimated maximum sales.

Step 5: Calculate and Estimate the Limit

Based on the data, sales seem to plateau around 50.8 after month 14. By visually extending the asymptote horizontally in the plot, you can estimate that total sales might stabilize around 55 to 60 (in thousands). This is the estimated total possible sales of the game.

Key Concepts

Sales Growth Trend

Understanding the sales growth trend is crucial for businesses when a new product hits the market. It provides insights into how well the product is being received by consumers. In general, the trend will start with a slow build as the product is introduced, followed by an acceleration as the product gains popularity.
As more consumers become aware of the product, sales often increase rapidly. This growth phase is exciting but it's also limited by factors such as market capacity and competition.
Eventually, sales will begin to slow, marking the start of the point of diminishing returns. By analyzing the sales growth trend, businesses can better understand the lifespan of a product's appeal.

• Helps identify early success or challenges in product sales.
• Aids in planning future marketing strategies.
• Alerts to upcoming shifts in consumer interest.

Scatter Plot Analysis

Utilizing scatter plot analysis provides a visual representation of the relationship between two variables: here, time (months) and sales. To analyze the data effectively, plot each point by placing one variable on the x-axis and the other on the y-axis. In this exercise, months are on the x-axis and sales in thousands on the y-axis.

This visual method is powerful as it makes patterns easier to spot. Specifically, we can observe how initially sales numbers increase rapidly, but over time, the rate of increase slows. This visual cue helps pinpoint significant changes such as the point of diminishing returns.

• Enables a clear view of data patterns over time.
• Highlights relationships between variables quickly.
• Helps in predicting future trends pragmatically.

Maximum Sales Prediction

Predicting maximum sales is about estimating a product's ultimate market potential. Once sales start to decrease in velocity after the point of diminishing returns, we use this data to project total possible sales. It's assumed that eventually, sales taper off into a steady state, forming what is known as a horizontal asymptote.

In our exercise, after analyzing the plot, sales seemed to level off around 50.8 after month 14. By extending this pattern, one could hypothesize the sales to stabilize between 55 to 60 in thousands by observing the horizontal progression of this asymptote. This provides an essential ceiling for expected salesop, helping businesses set realistic expectations and allocate resources effectively.

• Aids in financial planning by setting potential revenue ceilings.
• Provides benchmarks for sales teams and production units.
• Facilitates inventory and supply chain forecasting.