Question

Glorious Gadgets is a retailer of astronomy equipment. They purchase equipment from a supplier and then sell it to customers in their store. The function C(x)=4x+18000x⁻¹ +6000 models their total inventory costs (in dollars) as a function of x the lot size for each of their orders from the supplier. The inventory costs include such things as purchasing, processing, shipping, and storing the equipment. What lot size should Glorious Gadgets order to minimize their total inventorv costs? (NOTE: your answer must be the whole number that corresponds to the lowest cost.) What is their minimum total inventory cost?

Answer 1

To minimize Glorious Gadgets' total inventory costs, we find the derivative of the cost function, set it equal to zero, and solve for x. The resulting lot size that minimizes costs is rounded down to the nearest whole number. By substituting this value into the cost function, we find the minimum total inventory cost.

Step-by-step explanation:

To find the lot size that will minimize Glorious Gadgets' total inventory costs, we need to determine the minimum point of the cost function, C(x) = 4x + 18000x⁻¹ + 6000. This can be done by finding the derivative of the cost function, setting it equal to zero, and solving for x.

Taking the derivative of C(x) with respect to x:

C'(x) = 4 - 18000x⁻²

Setting C'(x) = 0 and solving for x:

4 - 18000x⁻² = 0

18000x⁻² = 4

x⁻² = 4/18000

x⁻² = 1/4500

1/x² = 1/4500

x² = 4500

x = √4500

x ≈ 67.08

Since the lot size must be a whole number, we round down to the nearest whole number.

Hence, the lot size that will minimize Glorious Gadgets' total inventory costs is 67.

To find the minimum total inventory cost, we substitute the lot size (x = 67) into the cost function:

C(67) = 4(67) + 18000(67⁻¹) + 6000

C(67) = 268 + 18000(1/67) + 6000

C(67) ≈ 417.91

Therefore, Glorious Gadgets should order a lot size of 67 to minimize their total inventory costs, and the minimum total inventory cost is approximately $417.91.