Questions: 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+56875 x^-1+22750 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?

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+56875 x^-1+22750 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?

Transcript text: 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 $C(x)=2.5 x+56875 x^{-1}+22750$ 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.) $\square$ What is their minimum total inventory cost? \$ $\square$

Solution

Solution Steps

Step 1: Find the critical point

To minimize the total inventory costs, we first find the critical point by setting the derivative of the cost function equal to zero: $C'(x) = a - bx^{-2} = 0$. Solving for $x$, we find $x = \sqrt{\frac{b}{a}} = 150.83$.

Step 2: Verify it's a minimum

To ensure this critical point is indeed a minimum, we check the second derivative: $C''(x) = 2bx^{-3}$. Since $C''(x) = 0.03 > 0$ for all $x > 0$, the critical point is a minimum.

Step 3: Calculate the minimum total inventory cost

Since the lot size must be a whole number, we evaluate the cost function at the floor and ceiling of the critical point: $x = 150$ and $x = 151$. The minimum cost is found to be $23504.16 at a lot size of 151.