Question

For the cost function C(x)=5000+66x+0.002x^3 find:
A) The production level that will minimize the average cost.
B) The minimal average cost.

Answer 1

what's x? are there other constraits? i don't see well where this comes from..

Answer 2

im not sure thats all the information they gave me.. so im clueless on whats the first step

Answer 3

Usually you want to minimize some function under some constraints.
Looking at the cost here, if we can take any \(x\) we want, we can go take negative values, because of the \(x^3\) in the function C, the cost will decrease a lot!..

Answer 4

maybe whaht they want is to look for the \(x\) minimizing \(C(x)\) , with values of \(x\) between 0 and some number A. Do you see such information in the info you have? Maybe A is found by thinking about a natural constraint.

Answer 5

to tell you the truth im lost i know that the weeks before we were finding marginal cost and i know you have to take the derivative but im not such fr this one

Answer 6

\(C'(x)=66+\frac{3}{500}x^2\), but this is never equal to zero. always positive..
hmm that would mean the cost is always increasing? can that be?

Answer 7

maybe....no i dont think so do you think it has something to do with revenue and price i know that revenue=px

Answer 8

Can you give all the info you have?

But, I think I will not be able to help much because it seems I haven't studied this topic, or a bit but I forgot..

Answer 9

thats all the information they gave me have you taken calculus before

Answer 10

Calculus yes, I thought this was some economy or optimization course..
It's not?
What annoys me is that I can't put a name (x? p?) to what you call "production level"..

Answer 11

yeah this is calculus.. its my first time taking it so im unsure my professor did not explain the sunject at all i was thinking that production was revenue- cost but i think that is profit

Answer 12

are you sure it's x^3 and not x^2 in the statement?

Answer 13

yeah i am sure i just doubled checked

Answer 14

ehm I found this: Average cost = C(x)/x.

Answer 15

You need to differentiate this and equalize to zero to find the x minimizing the average cost.
So, find x verifying: (C(x)/x)' = 0
which is (5000/x + 66 + x^2/500)' = 0
<=> -5000/x^2 + x/250 = 0
etc.

Answer 16

wait did you say to put C(x) divided by another x or is x another equation

Answer 17

so you found the derivative of c(x) right and then we set it equal to zero

Answer 18

You have to solve \(\left(\frac{C(x)}{x}\right)'=0\).

Answer 19

in this case what is x though is it just the variable x

Answer 20

\(\frac{C(x)}{x}=\frac{5000}{x}+66+\frac{x^2}{500}\).
So the derivative of this is \(-\frac{5000}{x^2}+0+\frac{2x}{500}\).

Answer 21

It seems \(x\) is the production level.. I'm not sure since this is kind of new for me too lol

Answer 22

so \(\frac{5000}{x^2} = \frac{2x}{500} \iff 5000\times 250 = x^3\iff 5\times 10^3 \times 5 \times 5 \times 10 = x^3\).
So \(x=\sqrt[3]{5^3 \times 10^3\times10} = 5\times 10\times \sqrt[3]{10}\). Does that seem reasonable?

Answer 23

oh okay it seem a little complicated but i just need to pay attention on what you did carefully

Answer 24

that is, this \(x=50\sqrt[3]{10}\) is the production level that minimizes the cost and to find the minimal average cost, just plug this value in the cost function\(C\).

Answer 25

What you need to remember is that given a cost function \(C(x)\), the average cost function is \(C(x)/x\) and the minimum production level is the solution of \((C(x)/x)'=0\). Afterwards it's just calculus (need to train that though...).

Answer 26

oh okay so i just plug this in for the x values in the cost function

Answer 27

yea, the answer to a) is \(50\sqrt[3]{10}\), the answer to b) is \(C(50\sqrt[3]{10})=...\).

Answer 28

for some reason its telling me that the answer does not have a cube root ??

Answer 29

do you know the answer or whow are you checking?

Answer 30

no i typed in the answer i have homework online and its not accepting it

Answer 31

maybe if i convert the bure root into a 1/3 it may work let me try it

Answer 32

nope still didnt let me :(

Answer 33

what if you type 107.72 ?

Answer 34

prod. level minimizing cost : 107.72..
minimum average cost: C(107.72) = 14609.63..