Let's solve the problem step by step: ### **Given:** The total inventory cost function: \[ C(z) = 1.5z + \frac{105{,}000}{z} + 3000 \] where \( z \) is the **lot size** for each order. We are to: 1. **Find the lot size \( z \) that minimizes the total inventory cost.** 2. **Find the minimum total inventory cost.** --- ## **Step 1: Find the critical points** To minimize the cost, we take the derivative of \( C(z) \) with respect to \( z \), set it to 0, and solve for \( z \): \[ C(z) = 1.5z + \frac{105{,}000}{z} + 3000 \] Compute the derivative: \[ C'(z) = 1.5 - \frac{105{,}000}{z^2} \] Set derivative equal to 0: \[ 1.5 - \frac{105{,}000}{z^2} = 0 \] \[ 1.5 = \frac{105{,}000}{z^2} \] Multiply both sides by \( z^2 \): \[ 1.5z^2 = 105{,}000 \] Solve for \( z^2 \): \[ z^2 = \frac{105{,}000}{1.5} \] \[ z^2 = 70{,}000 \] \[ z = \sqrt{70{,}000} \] \[ z \approx 264.575 \] --- ## **Step 2: Whole Number Lot Size** The problem specifies to use the whole number that corresponds to the **lowest cost**. Let's check \( z = 264 \) and \( z = 265 \). --- ## **Step 3: Evaluate Costs** ### For \( z = 264 \): \[ C(264) = 1.5(264) + \frac{105,000}{264} + 3000 \] \[ = 396 + 397.727 + 3000 \] \[ = 3793.727 \] ### For \( z = 265 \): \[ C(265) = 1.5(265) + \frac{105,000}{265} + 3000 \] \[ = 397.5 + 396.226 + 3000 \] \[ = 3793.726 \] \( 3793.726 < 3793.727 \) So, \( z = 265 \) gives the lowest cost. --- ## **Step 4: Final Answers** ### **1. Lot size that minimizes total inventory cost:** \[ \boxed{265} \] ### **2. Minimum total inventory cost:** \[ \boxed{3793.73} \] (Rounded to two decimal places) --- ### **Summary Table** | Lot Size (\( z \)) | Total Inventory Cost (\( C(z) \)) | |--------------------|-----------------------------------| | 264 | 3793.73 | | 265 | 3793.73 | --- **Final Answers:** - **Lot size:** \( \boxed{265} \) - **Minimum total inventory cost:** \( \boxed{3793.73} \)

The optimal lot size is found by differentiating the total cost function and setting it to zero, which identifies the point where costs are minimized. The minimum total inventory cost is obtained by evaluating the cost at the optimal lot size.