Let's analyze the problem step by step using proper formulas. --- # Step 1: Understand the initial data - **Initial number of engines per truck:** \(N_{initial}\) (not given, but assume it is more than 100) - **Reduced number of engines per truck:** \(N_{reduced} = 100\) - **Cost trip:** \$,000 - **Number of trips:** \(T\ (assumed to be the same for both cases) - **Total cost for each truck:** \(C = \text{cost per trip} \times T\) --- # Step 2: Calculate the change in inventory costs ### Impact on inventory costs: - **Before reduction:** Inventory costs depend on the number of engines per truck, say \(C_{inventory, initial}\) - **After reduction:** Inventory costs depend on the new number \(N_{reduced} = 100\) Since the problem asks for the **impact on annual inventory costs**, and it's related to the number of engines loaded, an assumption is that inventory costs are proportional to the number of engines per truck. \[ \text{Change in inventory costs} \propto N_{initial} - 100 \] --- # Step 3: Calculate the total savings or increase in costs ### **Cost savings due to fewer engines:** The reduction from \(N_{initial}\) engines to 100 engines reduces inventory costs proportionally. ### **Cost implications:** - Each trip costs \$1,000 - The number of trips \(T\) per year (assumed to be the same in both cases) Total annual cost: \[ \text{Total cost} = \text{Cost per trip} \times T \] --- # Step 4: Determine the effect on inventory costs Assuming the inventory cost per engine per trip is proportional to the number of engines, the change in inventory costs: \[ \Delta C_{inventory} = \text{Cost per engine} \times (N_{initial} - 100) \] But without specific data on per-engine inventory costs or initial number of engines, we can't compute a precise numeric answer. --- # Step 5: Find the optimal number of engines if load is 100 engines The problem states that the load of 100 engines is optimal, implying that at this load: - **Total cost per truck per trip:** \(\$1,000\) - **Total annual inventory costs:** minimized Thus, the **cost of each truck** if load is 100 engines, including inventory costs, is: \[ C_{total} = C_{operation} + C_{inventory} \] where: - \(C_{operation} = \$1,000 \times T\) - \(C_{inventory}\) is proportional to the number of engines loaded (here, 100 engines) --- # Final note: Without specific data on initial number of engines, per-engine inventory costs, or total trips, the best we can do is outline the proportional relationships and the impact of reducing the number of engines on inventory costs. --- **Summary:** - The reduction in engines from \(N_{initial}\) to 100 decreases inventory costs proportionally. - The total cost per truck per year is primarily affected by the trip cost (\$1,000 per trip) and inventory costs. - Optimal load at 100 engines minimizes total costs, balancing operational costs and inventory costs. --- If additional data (like initial number of engines or per-engine costs) is provided, a more precise numerical solution can be derived.