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)=4.5x+42000x −1

+14000 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?

Answer 1

To minimize the function C(x) = 4.5x + 42000/x + 14000, calculate the first derivative, set it to zero and solve for x to find the lot size that minimizes the total inventory cost. Plug this lot size into the cost function to find the minimum cost, making sure to round to a whole number.

Step-by-step explanation:

The question asks what lot size Glorious Gadgets should order to minimize their total inventory costs, given the function C(x) = 4.5x + 42000/x + 14000. This question involves finding the minimum of a cost function, which is a common problem in calculus related to optimization. To minimize the cost function, we must find the derivative of C(x) and set it equal to zero to find the critical points. However, it seems there's a typo in the function provided as '42000x' does not make sense with '−1' immediately following it without any operation between them. Assuming the correct function is C(x) = 4.5x + 42000/x + 14000, we would proceed as follows:

Firstly, find the first derivative of C(x) with respect to x:

C'(x) = 4.5 - 42000/x2

Then set the derivative equal to zero and solve for x:

4.5 - 42000/x2 = 0

x2 = 42000 / 4.5

x = sqrt(42000 / 4.5)

Once we have the value of x, we check to make sure this x value gives us a minimum on the cost curve. Then, to find the minimum total inventory cost for Glorious Gadgets, we plug the value of x back into the original cost function. The resulting value would be their minimum total inventory cost. The final answers should be provided as whole numbers.

Answer 2

To minimize total inventory costs, Glorious Gadgets should order a lot size of 9333 units. The minimum total inventory cost is $42,027,447.67.

Step-by-step explanation:

To find the lot size that will minimize Glorious Gadgets' total inventory costs, we need to find the x-value that corresponds to the minimum value of the function C(x) = 4.5x + 42000x -1 + 14000. This can be done by taking the derivative of the function and setting it equal to zero:

C'(x) = 4.5 - 42000 = 0

Now solve for x:

4.5x = 42000

x = 42000/4.5 = 9333.33

Rounding to the nearest whole number, the optimal lot size for Glorious Gadgets to order is 9333 units.

To find the minimum total inventory cost, substitute the optimal lot size into the function:

C(9333) = 4.5(9333) + 42000(9333) -1 + 14000 = $42,027,447.67