Question

section 2.10: applied optimization (homework)
score: 140/150 answered: 14/15
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question 15
0/10 pts 4 99 details
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 $c(x)=4.5x + 18750x^{-1}+7500$ 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?

Explanation:

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

We have \(C(x)=4.5x + 18750x^{-1}+7500\). Using the power - rule \((x^n)^\prime=nx^{n - 1}\), the derivative \(C^\prime(x)=4.5-18750x^{-2}\).

Step2: Set the derivative equal to zero

Set \(C^\prime(x) = 0\), so \(4.5-18750x^{-2}=0\). Then \(4.5=\frac{18750}{x^{2}}\). Cross - multiply to get \(4.5x^{2}=18750\).

Step3: Solve for \(x\)

Divide both sides of \(4.5x^{2}=18750\) by \(4.5\): \(x^{2}=\frac{18750}{4.5}=\frac{187500}{45}=\frac{12500}{3}\). Then \(x=\sqrt{\frac{12500}{3}}\approx64.55\). Since \(x\) must be a whole number, we consider \(x = 64\) and \(x = 65\).

Step4: Calculate \(C(x)\) for \(x = 64\) and \(x = 65\)

\(C(64)=4.5\times64 + 18750\times64^{-1}+7500=4.5\times64+\frac{18750}{64}+7500=288 + 292.97+7500=8080.97\).
\(C(65)=4.5\times65 + 18750\times65^{-1}+7500=292.5+\frac{18750}{65}+7500=292.5 + 288.46+7500=8080.96\).

Answer:

Lot size: 65
Minimum cost: 8080.96