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? $ 12620 question help: video > next 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)=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? $ 12620 question help: video > next question

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}).

Step2: Set the derivative equal to zero

Set (C^\prime(x) = 0), so (4-\frac{24000}{x^{2}}=0). Add (\frac{24000}{x^{2}}) to both sides: (4=\frac{24000}{x^{2}}). Cross - multiply: (4x^{2}=24000). Divide both sides by 4: (x^{2}=6000). Take the square root: (x=\sqrt{6000}\approx77.46).

Step3: Check the second - derivative

The second - derivative (C^{\prime\prime}(x)=48000x^{-3}=\frac{48000}{x^{3}}). When (x > 0), (C^{\prime\prime}(x)>0), so the function is concave up at the critical point, and the critical point is a minimum. Since (x) must be a whole number, we check (x = 77) and (x = 78). For (x = 77): (C(77)=4\times77+24000\times77^{-1}+12000) (=308+\frac{24000}{77}+12000) (=308 + 311.69+12000\approx12619.69). For (x = 78): (C(78)=4\times78+24000\times78^{-1}+12000) (=312+\frac{24000}{78}+12000) (=312+307.69+12000\approx12619.69).

Answer:

The lot size (x = 77) or (x = 78) (either whole number gives the minimum cost). The minimum total inventory cost is approximately ($12620).