\begin{tabular}{l} the trektonics store begins each month with 1150 phasers in stock. \\ this stockis depleted each month and reordered. The carrying cost per phasers is \( \$ 36 \) per year \\ and the fixed order cost is \( \$ 560 \). \\ \hline\end{tabular} \begin{tabular}{l} calculate the optimal number of orders per year \\ \hline\end{tabular}

To tackle this problem, we can use the Economic Order Quantity (EOQ) model. First, we need to determine the demand for the phasers. Assuming the store sells all 1150 phasers by the end of each month, the annual demand (D) would be \( 1150 \text{ phasers/month} \times 12 \text{ months} = 13,800 \text{ phasers/year} \). Next, we need to find the optimal order quantity (Q). The formula for EOQ is: \[ Q^* = \sqrt{\frac{2DS}{H}} \] where \( D \) is the annual demand, \( S \) is the fixed order cost, and \( H \) is the carrying cost per unit per year. Plugging in the values: \[ D = 13,800 \text{ phasers/year} \\ S = 560 \\ H = 36 \] \[ Q^* = \sqrt{\frac{2 \times 13,800 \times 560}{36}} \\ Q^* = \sqrt{\frac{15,424,800}{36}} \\ Q^* \approx \sqrt{428,000} \\ Q^* \approx 655.74 \] We round this to the nearest whole number, so the optimal order quantity (Q) is 656 phasers. Now, we calculate the optimal number of orders per year (N): \[ N = \frac{D}{Q^*} = \frac{13,800}{656} \approx 21.04 \] Rounding down, the optimal number of orders per year is approximately 21. So, the optimal number of orders per year is **21**.