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?

i cant help

To determine the number of times per year and the lot size the store should order pool tables to minimize its inventory costs, we can use the derivative method. Let's break down the problem into smaller steps:

Step 1: Define the variables:
Let's assume:
- "C" represents the total cost of inventory (storage cost + delivery cost)
- "n" represents the number of orders per year
- "Q" represents the lot size (number of pool tables per order)
- "C1" represents the storage cost per pool table per year ($20)
- "C2" represents the delivery cost per order ($40)
- "D" represents the demand for pool tables per year (100)

Step 2: Express the cost function:
The annual cost can be determined by multiplying the cost per pool table by the number of pool tables per year, and adding the delivery cost:
C = n * C1 * Q + (D / n) * C2

Step 3: Calculate the derivative of the cost function:
To optimize the cost function, we differentiate it with respect to "n" while treating "Q" as a constant:
dC/dn = C1 * Q - (D / n^2) * C2

Step 4: Set the derivative equal to zero and solve for "n":
Setting dC/dn = 0, we have:
C1 * Q - (D / n^2) * C2 = 0
C1 * Q = (D / n^2) * C2
n^2 = (D * C2) / (C1 * Q)
n = √((D * C2) / (C1 * Q))

Step 5: Find the value of "Q":
To find the optimal lot size, we substitute the value of "n" calculated in Step 4 back into the derived equation of the annual cost:
Q = D / n

Step 6: Calculate the values:
Now we substitute the given values into the equations:
C1 = $20
C2 = $40
D = 100

Using these values, we can calculate "n":
n = √((100 * 40) / (20 * Q))
n = √(4000 / (20 * Q))
n = √(200 / Q)
n = √(200Q^(-1))

And then we can calculate the lot size "Q":
Q = 100 / n

Step 7: Determine the minimum cost:
To find the minimum cost, we substitute the optimal values of "n" and "Q" back into the cost equation:
C = n * C1 * Q + (D / n) * C2

By following these steps, you can find the number of times per year and the lot size the store should order pool tables to minimize inventory costs.