glorious gadgets is a retailer of astronomy equipment. they purchase equipment from a supplier and then sell…

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)=4x + 24000x^{-1}+12000 ) 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)=4x + 24000x^{-1}+12000 ) 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

Explanation:

Step1: Find the derivative of (C(x))

The function is (C(x)=4x + 24000x^{-1}+12000). Using the power rule ((x^n)^\prime=nx^{n - 1}), the derivative (C^\prime(x)=4-24000x^{-2}). Set (C^\prime(x) = 0), so (4-\frac{24000}{x^{2}}=0).

Step2: Solve for (x)

From (4-\frac{24000}{x^{2}}=0), we can rewrite it as (\frac{24000}{x^{2}}=4). Then (x^{2}=\frac{24000}{4}=6000), and (x=\sqrt{6000}\approx77.46). We also need to check the second - derivative (C^{\prime\prime}(x)=\frac{48000}{x^{3}}). When (x>0), (C^{\prime\prime}(x)>0), so the function is concave up at the critical point. Since (x) represents the lot size (a whole number), we check (x = 77) and (x = 78). (C(77)=4\times77+\frac{24000}{77}+12000\approx308 + 311.69+12000=12619.69) (C(78)=4\times78+\frac{24000}{78}+12000\approx312+307.69 + 12000=12619.69)

Answer:

The lot size (x = 77) or (x = 78) (both give the same cost). The minimum total inventory cost is ($12619.69\approx$12620)