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 $\(\boxed{32293.73}\).