# Glorious Gadgets is a retailer of astronomy equipment. They purchase equipment from a supplier...

## 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 {eq}C(x) = 1x + 120,000x^{-1} + 40,000 {/eq} models their total inventory costs (in dollars) as a function of {eq}x {/eq} 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?

## Differentiation:

Differentiation is used to convert a whole into parts. The first derivative of any function gives us an equation for finding the slope of a tangent. Differentiation of constant is always zero.

Solution:

Given

{eq}C(x) = 1x + 120000{x^{ - 1}} + 40000 {/eq}

C(x) is the total inventory cost

x is the lot size for each of their orders from the supplier

First we differentitate the function C(x) with respect to x

So

{eq}\dfrac{d}{{dx}}C(x) = 1 + ( - 1) \times 120000{x^{ - 2}} + 0 {/eq}

Now we let left hand side term equal to zero to minimize total inventory cost

So

{eq}\dfrac{d}{{dx}}C(x) = 1 + ( - 1) \times 120000{x^{ - 2}} {/eq}

Becomes

{eq}0 = 1 + ( - 1) \times 120000{x^{ - 2}} {/eq}

Solving for x gives

{eq}{x^2} = \dfrac{{120000}}{1} = 120000 {/eq}

So

{eq}x = \sqrt {120000} = 346 {/eq}

Now the minimum total inventory cost is obtained by putting value of x in the function C(x)

So

{eq}C(346) = 1(346) + 120000{(346)^{ - 1}} + 120000 {/eq}

Which gives

{eq}\begin{align*} C(346) &= (346) + \dfrac{{120000}}{{346}} + 120000\\ C(346) &= 346 + 346 + 120000\\ &= 120692 \end{align*} {/eq}