Problem

The following table shows the total sales, in thousands, since a new game was brought to market. \begin{tabular}{c|c|c|c|c|c|c|c|c} \hline Month & 0 & 2 & 4 & 6 & 8 & 10 & 12 & 14 \\ \hline Sales & 0 & 2.4 & 5.6 & 8.0 & 9.2 & 9.6 & 9.9 & 10.0 \\ \hline \end{tabular} (a) Plot this data and determine the point of diminishing returns. Enter the closest value in the table. The point of diminishing returns occurs months after the game is introduced. (b) Predict total possible sales of this game, using the point of diminishing returns from the table. Total sales $\approx$

Solution

Step 1 :From the given data, we can observe that the sales increase from 0 to 2.4 in the first 2 months, then from 2.4 to 5.6 in the next 2 months, then from 5.6 to 8.0 in the next 2 months, then from 8.0 to 9.2 in the next 2 months, then from 9.2 to 9.6 in the next 2 months, and finally from 9.6 to 9.9 in the next 2 months.

Step 2 :The rate of increase in sales is decreasing over time, which indicates that the point of diminishing returns is being reached.

Step 3 :The point at which the rate of increase begins to decrease significantly is between month 6 and month 8. Therefore, the point of diminishing returns occurs 8 months after the game is introduced.

Step 4 :The total possible sales of this game can be estimated using the point of diminishing returns. After the point of diminishing returns, the sales increase at a decreasing rate.

Step 5 :From the table, the sales at the point of diminishing returns (month 8) is 9.2. The sales increase after the point of diminishing returns is 0.8 (from 9.2 to 10.0).

Step 6 :Therefore, the total possible sales of this game is approximately 10.0 (in thousands).

Step 7 :So, the point of diminishing returns occurs 8 months after the game is introduced and the total possible sales of this game is approximately \(\boxed{10.0}\) (in thousands).

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Source: https://solvelyapp.com/problems/t3jT2TtP85/

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