The need of a vegetable store in apples is $10,000 \, \text{kg}$ per quarter, and apples are consumed evenly and continuously. Apples are ordered once a quarter and delivered in batches of the same volume specified in the order. Storage of one kg of apples in the warehouse costs $20 \, \text{kopecks}$ per day, and delivery of a batch costs $12,000 \, \text{rubles}$ . Delay in selling apples due to their unavailability is inadmissible. Determine the most economical batch volume and the interval between deliveries to be specified in the order (it is assumed that the supplier does not allow delays in deliveries).

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Q. The need of a vegetable store in apples is

10,000 \, \text{kg}

per quarter, and apples are consumed evenly and continuously. Apples are ordered once a quarter and delivered in batches of the same volume specified in the order. Storage of one kg of apples in the warehouse costs

20 \, \text{kopecks}

per day, and delivery of a batch costs

12,000 \, \text{rubles}

. Delay in selling apples due to their unavailability is inadmissible. Determine the most economical batch volume and the interval between deliveries to be specified in the order (it is assumed that the supplier does not allow delays in deliveries).

Define Variables and Problem: Define the variables and the problem. $\newline$ Let $Q$ be the batch volume in kilograms, and let $T$ be the interval between deliveries in days. The store needs $10,000$ kg of apples per quarter ( $90$ days). The cost of storage is $0.20$ rubles per kg per day, and the delivery cost for each batch is $12,000$ rubles.
Calculate Deliveries per Quarter: Calculate the number of deliveries per quarter. $\newline$ Since the store orders once a quarter and needs $10,000$ kg of apples, the number of deliveries per quarter is $10,000 / Q$ .
Calculate Storage Cost: Calculate the total storage cost per quarter. The average inventory level is $\frac{Q}{2}$ , because the store starts with $Q$ kg and ends with $0$ kg before the next delivery. The storage cost per quarter is $\left(\frac{Q}{2}\right) * 0.20 \text{ rubles/kg/day} * 90 \text{ days}.$
Calculate Delivery Cost: Calculate the total delivery cost per quarter. The delivery cost per quarter is $(10,000 / Q) \times 12,000$ rubles, because each batch costs $12,000$ rubles and there are $10,000 / Q$ batches per quarter.
Write Total Cost Function: Write the total cost function. $\newline$ The total cost per quarter, $C(Q)$ , is the sum of the storage cost and the delivery cost. So, $C(Q) = (\frac{Q}{2}) \times 0.20 \times 90 + (\frac{10,000}{Q}) \times 12,000$ .
Find Derivative of Cost Function: Find the derivative of the total cost function with respect to $Q$ . To minimize the total cost, we take the derivative of $C(Q)$ with respect to $Q$ and set it to zero. The derivative is $C'(Q) = (0.20 \times 90 / 2) - (10,000 \times 12,000 / Q^2)$ .
Solve for Q: Solve for Q. $\newline$ Setting the derivative equal to zero gives us $(0.20 \times 90 / 2) = (10,000 \times 12,000 / Q^2)$ . Solving for Q, we get $Q^2 = (10,000 \times 12,000) / (0.20 \times 90 / 2)$ . Let's calculate this. $\newline$ $Q^2 = (10,000 \times 12,000) / (0.20 \times 90 / 2)$ $\newline$ $Q^2 = (120,000,000) / (9)$ $\newline$ $Q^2 = 13,333,333.33$ $\newline$ $Q = \sqrt{13,333,333.33}$ $\newline$ $Q \approx 3650$ kg
Calculate Interval Between Deliveries: Calculate the interval between deliveries, $T$ . Since the store consumes apples continuously, $T$ is the time it takes to sell the batch volume $Q$ . The store needs $10,000$ kg per $90$ days, so the daily consumption is $10,000 / 90$ kg/day. The interval $T$ is then $Q$ divided by the daily consumption. $T = \frac{Q}{(10,000 / 90)}$ $T = \frac{3650}{(10,000 / 90)}$ $T = \frac{3650}{(111.11)}$ $T \approx 32.85$ days