A sporting goods store sells 100 pool tables per year. It costs $20 to store one pool table for a year, based on the average inventory on hand. It costs $40 for each delivery of pool tables. How many times per year and in what lot size should the store order pool tables to minimize its inventory costs?

derivative way, help please?

To find the optimal order quantity and frequency for the pool tables, we need to minimize the total inventory costs. The inventory costs consist of the carrying costs and the ordering costs.

Let's assume the order quantity is 'q' (in terms of pool tables) and the order frequency is 'f' (per year).

The carrying cost is given by $20 per pool table per year, and since the average inventory on hand is 'q/2' (assuming continuous replenishment), the carrying cost is ($20 * q/2) = $10q.

The ordering cost is $40 per order, and since there are '100' pool tables sold per year, the order frequency is (100/q) orders per year. Therefore, the ordering cost is ($40 * 100/q) = $4000/q.

The total inventory cost is the sum of the carrying cost and the ordering cost:

Total Cost = Carrying Cost + Ordering Cost
Total Cost = $10q + $4000/q

To find the optimal order quantity and frequency, we can take the derivative of the total cost function with respect to 'q', set it equal to zero, and solve for 'q':

d(Total Cost)/dq = 10 - 4000/q^2 = 0

Simplifying the equation, we get:

10 = 4000/q^2

q^2 = 4000/10

q^2 = 400

Taking the square root of both sides, we find:

q ≈ 20

Therefore, the optimal order quantity would be approximately 20 pool tables.

To determine the order frequency, we divide the total annual demand (100 pool tables) by the order quantity:

f = 100/q ≈ 100/20 = 5

Therefore, the optimal order frequency would be 5 orders per year, with each order consisting of approximately 20 pool tables.

Keep in mind that these are approximate values based on the assumptions made, and you may need to consider other factors or constraints in the actual decision-making process.