Problem statement: Product's storing costs depend on the amount of product ordered and that is represented by a function K(x)=(30000/x)+5x. What is the optimal amount of product ordered in order to minimize costs. Find out by determining the min of that function using derivation.
Solution: Taking a derivative of the function i get (-30000/x^2)+5. Making the derivative equal 0 and solving for x gives me an answer of 20sqrt(15) and then plugging that back into the k(x) function i get an answer of 775 units of product. Am i right?
Thanks
I think some ideas got confused in the course of discussion. Let me make some suggestions.
First, the problem really should have been clearer about some details. Here is what I might expect it to say (making some assumptions):
A product's storage cost depends on the amount of product ordered and is given by the function K(x)=(30000/x)+5x, where x is the quantity of product (in kilograms), and K is the cost of storing that quantity. What is the optimal amount of product ordered in order to minimize cost (to the nearest tenth of a kilogram)? Find out by determining the minimum of that function using the derivative.
Note a couple things. Others have assumed the quantity must be an integer, because it is a number of items. I've supposed it is measured in some
continuous unit (an amount of product rather than a number of products), which would change things. (I suspect that in the original language this might have been clearer.) Also, the problem as you stated it didn't
define the variables; that's a bad thing. I've made it clearer both that x is the amount of product and that K is the cost for that entire amount,
not per kilogram or per item (as is done in some problems of this sort). That ambiguity hasn't been mentioned yet. Finally, if it isn't clear that x is an integer from its definition, something should be said about the form required; on my supposition, it's necessary.
In post #3, you added, "K(x) represents costs to warehouse said
units of the product." Assuming this was either stated or clearly implied in the problem, it suggests that my supposition above is wrong, and x is in fact (as everyone has assumed) a discrete number of items. So let's rewrite the problem that way. (I made the opposite assumption entirely for the sake of showing the issue!)
A product's storage cost depends on the number of items ordered and is given by the function K(x)=(30000/x)+5x, where x is the quantity of product, and K is the cost of storing that quantity. What is the optimal amount of product ordered in order to minimize cost? Find out by determining the minimum of that function using the derivative.
In this case, the answer should be a specific integer; you would round up or down and see which produces the lower cost.
Finally, you stated at the end, "plugging that back into the k(x) function i get an answer of 775 units of product". I hope you see by now that that would be wrong if K(x) gives the cost; you were asked for the quantity, x, not K(x). But then in post #8 you said, "I have made a mistake. Cost indeed is not relevant, but the function K(x) represents the quantity of product. sorry bout that, I should have re-read my original post before commenting." I think you said this because of a misunderstanding of the questions being asked. The problem clearly says that K(x) gives the cost, and surely implies that x is the quantity. Cost is not irrelevant (being what is to be minimized), but it is not what is asked for. (Unless perhaps you didn't translate that part of the problem correctly.)
I hope things are clear now. Putting both 77 and 78 into the function, we find that K(77) = 774.61039 and K(78) = 774.61538, so 78 is technically the correct answer, but, indeed, it would be wrong to count 77 as incorrect, with such a tiny difference.
