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 $C(x)=2.5 x+63000 x^{-1}+31500$ 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 inventory costs? (NOTE: your answer must be the whole number that corresponds to the lowest cost.)
What is their minimum total inventory cost? S
Answer
Final Answer: The lot size that Glorious Gadgets should order to minimize their total inventory costs is \(\boxed{159}\). The minimum total inventory cost is approximately $\(\boxed{32293.73}\).
Step 1 ：The function \(C(x)=2.5 x+63000 x^{-1}+31500\) models the total inventory costs (in dollars) as a function of \(x\), the lot size for each of their orders from the supplier.
Step 2 ：To find the minimum of this function, we first find its derivative and set it equal to zero. The derivative of \(C(x)\) is \(C'(x) = 2.5 - 63000/x^2\).
Step 3 ：Solving \(C'(x) = 0\) gives us two critical points, approximately -158.745078663875 and 158.745078663875.
Step 4 ：We then check the second derivative at these points to confirm that they are minima. The second derivative of \(C(x)\) is \(C''(x) = 126000/x^3\).
Step 5 ：The critical point that gives a minimum is approximately 158.745078663875. However, the problem asks for a whole number that corresponds to the lowest cost. Therefore, we need to check the integer values on either side of this number (158 and 159) to see which gives the lower cost.
Step 6 ：Substituting \(x = 158\) into \(C(x)\) gives a cost of approximately 32293.7341772152.
Step 7 ：Substituting \(x = 159\) into \(C(x)\) gives a cost of approximately 32293.7264150943.
Step 8 ：Comparing these two costs, we find that the minimum cost is approximately 32293.7264150943, which corresponds to a lot size of 159.
Step 9 ：Final Answer: The lot size that Glorious Gadgets should order to minimize their total inventory costs is \(\boxed{159}\). The minimum total inventory cost is approximately $\(\boxed{32293.73}\).
