The annual inventory cost for a manufacturer is given by
C = 1,008,000/Q + 6.5Q
where Q is the order size when the inventory is replenished. Find the change in annual cost when Q is increased from 310 to 311. (Round your answer to two decimal places.)


To find the change in the annual inventory cost when \( Q \) increases from 310 to 311, we can calculate the cost at both values of \( Q \) and find the difference.

The formula for the annual inventory cost is given by:

\[
C(Q) = \frac{1,008,000}{Q} + 6.5Q
\]

### Step 1: Calculate the cost at \( Q = 310 \)
\[
C(310) = \frac{1,008,000}{310} + 6.5 \times 310
\]

### Step 2: Calculate the cost at \( Q = 311 \)
\[
C(311) = \frac{1,008,000}{311} + 6.5 \times 311
\]

### Step 3: Find the difference in cost
The change in cost is:
\[
\Delta C = C(311) - C(310)
\]

Let's compute these values.

I will calculate this now.The change in annual inventory cost when \( Q \) increases from 310 to 311 is approximately \(-3.96\). This indicates a decrease in cost of about 3.96 units.

Would you like more details or further clarification?

Here are some related questions:
1. What happens to the cost if \( Q \) increases to 320 instead of 311?
2. How can we minimize the annual inventory cost using this formula?
3. What is the effect of increasing the fixed cost on the total cost?
4. What does the term \( \frac{1,008,000}{Q} \) represent in this context?
5. How would the result change if the variable cost was higher, say \( 10 \) instead of \( 6.5 \)?

**Tip:** For optimization problems, differentiating the cost function and finding critical points helps identify minimum or maximum values.

The annual inventory cost for a manufacturer is given by C = 1,008,000/Q + 6.5Q where Q is the order size when the inventory is replenished. Find the change in annual cost when Q is increased from 310 to 311. (Round your answer to two decimal places.)

Find the change in annual inventory cost using the cost function C = 1,008,000/Q + 6.5Q when order size increases from 310 to 311. This problem involves key calculus concepts such as cost functions and optimization, suitable for students in Grades 10-12.

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