Let's go through the questions in your worksheet.

Question 1:

The dashed line represents the average inventory for the three-month period.

Reasoning:

• Since the inventory decreases at a constant rate (as shown by the straight line), the average inventory level is the midpoint between the starting and ending inventory levels.
• The starting inventory is 500 units, and it decreases to 0 at the end of 3 months. The midpoint occurs at 1.5 months, and at that point, the inventory level is 250 units.
• Answer to (a): The average inventory is 250 units, as this is the midpoint.
• Answer to (b): The triangular area of the inventory (0 to 3 months) represents the total inventory over time, and a rectangle with a height of 250 would have the same area as this triangle. Therefore, 250 is the average inventory.
• Answer to (c): Both (a) and (b) provide valid explanations.

Question 2:

You are asked to find the average inventory if the quantity per order is $Q$.

• The average inventory in such scenarios is generally half the order size because the inventory decreases linearly from $Q$ to 0, similar to the graph shown in the first question.
• The formula for average inventory is: $\text{Average Inventory} = \frac{Q}{2}$
• Thus, the answer is (c) $Q/2$.

Would you like a detailed explanation of the inventory model? Here are five related questions:

1. What happens to the average inventory if the rate of sales increases?
2. How would the graph change if we reordered at 100 units instead of waiting for 0?
3. Can you derive the total inventory over time using integration for the given triangular graph?
4. What is the relationship between reorder frequency and average inventory?
5. How would the graph look if sales were not constant but accelerated over time?

Tip: The average inventory in many scenarios with constant depletion is simply half of the order size because the inventory decreases uniformly over time.