Question

I'm crying in tears here... I just need help with this.. why won't anybody help me???
Find the number of units x that produces the minimum average cost per unit C(with line over the C) in the given equation.
C = 0.001x^3 + 5x + 2

Answer 1

looks there is a typo in your equation, it should be :
C = 0.001x^\(\color{red}{2}\) + 5x + 2

can you please check once ? :)

Answer 2

No, it's x^3.. :(

Answer 3

it would be easier if it was x^2... I just don't get this cost stuff and the c with a line over it.. it doesn't make sense to me

Answer 4

Oh yeah they want to minimize the average cost \(\large \overline{C}\)

Answer 5

the example in the book has a weird derivative all over x.. i don't know why it does that for this type of problem.. why not just take the derivative and find critical points?

Answer 6

yeah.. i didn't know how to do that symbol in this website but yeah, thats what it's looking for

Answer 7

can you please take a screenshot of example problem and attach ?

Answer 8

yeah

Answer 9

oh, hold on for example problem.. that was the problem im trying to solve.. :/

Answer 10

no its okay :) i have understood hte problem

Answer 11

lets try this :
suppose you went for shopping and purchased 100 icecreams,
and lets say you had spent $1000 for those icecreams
can you tell me the cost of each icecream ?

Answer 12

each one cost $10.00

Answer 13

yes, but how did u get $10.00 ?

Answer 14

i divided

Answer 15

100 icecreams = $1000
so, 1 icecream = $1000/100

Answer 16

good :)
if u had purchased "x" icecreams,
and if it costs you "n" dollars,
can you tell me the cost for each icecream ?

Answer 17

umm, x/n?

Answer 18

think again

Answer 19

n/x?

Answer 20

sure ?

Answer 21

no...

Answer 22

n/x is correct,
to be sure, visualize a scenario :
x = 2 icecreams
n = 10 dollars

whats the cost of each icecream ?

Answer 23

is it 2/10
or 10/2


?

Answer 24

letters throw me off.. but i see it as 10/2

Answer 25

they throw everyone off haha
we just need to get use to them by using them more :)

Answer 26

right.. :)

Answer 27

okay so one more final question on icecreams to you :

supppose you purchased "x" icecreams,
and it costed you "0.001x^3 + 5x + 2" dollars

whats the cost of each icecream ?

Answer 28

(0.001x^3+5x+2)/x

Answer 29

what does that expression represent ?

Answer 30

cost of icecream?

Answer 31

thats the cost per icecream, right ?

Answer 32

yes both are same

Answer 33

instead of icecreams,
if we say "x" represents some other object/unit,
then your expression represents "cost per unit" right ?

Answer 34

yeah

Answer 35

i hope you see why we're dividing "x" when we're asked to find `average cost per unit`

Answer 36

so the c with line over it means average cost per unit..

Answer 37

Exactly !

\[\text{Average cost per item}(\overline{C}) = \dfrac{\text{Total cost of items}}{\text{number of items}}\]

Answer 38

so how do i solve?..

Answer 39

divide the given expression by "x"
then optimize it as usual

Answer 40

you know how to optimize a funciton, right ?

Answer 41

umm... no.. :(

Answer 42

okay its easy if you know derivatives
the method is same for optimizing ANY function

Answer 43

whats our function to be optimized ?

Answer 44

so the whole thing would be C= .001x^3+5x+2/x
So would C' be .001x^2+5+(2/x)?

Answer 45

exactly !

Answer 46

step1 : find the derivative, set it equal to 0, solve x

Answer 47

i'm a little shaky on that.. and then to find x is harder for me.. :/

Answer 48

function to be optimized : \(\large f(x) = .001x^2+5+(2/x)\)

Answer 49

find the derivative :
\(\large f'(x) = ?\)

Answer 50

.... oh man.. these fractions.. how do i change fractions into something easier?

Answer 51

\(\large 2/x = 2x^{-1}\)

Answer 52

would it be = 0.002x-2/x^2?

Answer 53

\[\large f(x) = .001x^2+5+(2/x)\]

\[\large
\begin{align}

f'(x) &= .001(2x)+0+2(-1/x^2) \\
&= .002x - 2/x^2

\end{align}
\]

Answer 54

Yep !

set it equal to 0 and solve x

Answer 55

0.002x-2/x^2=0.. solving for x.. whats best way to do this? i always just plug in random numbers until i get close to 0 if i can.. but i don't know if thats really best way to go about it..

Answer 56

the kings way of solving it is to get rid off the fraction first :
multiply x^2 both sides

Answer 57

\[\large
\begin{align}

f'(x) &= 0 \\
.002x - 2/x^2 &=0\\

0.002x^3 - 2 &=0

\end{align}
\]

Answer 58

I have just mulitplied x^2 both sides,
is that clear so far ?

Answer 59

yeah, but how come the .002 is x^3 now? do you have to multiply every term by x^2? because i would have just multiplied x^2 by x^2 and then x^2 by 0..

Answer 60

yes, instead of describing u how it works,
let me show u more steps, then it will be clear :


\[\large 0.002x - \dfrac{2}{x^2} = 0\]

multiply x^2 both sides :

\[\large x^2*\left(0.002x - \dfrac{2}{x^2}\right) = x^2*(0)\]

Answer 61

gotcha :)

Answer 62

good :) so what do u get for x after u solve ?

Answer 63

i divide both sides by .002 right?

Answer 64

actually i don't like decimals, so i would multiply by 1000 first

Answer 65

\[\large 0.002x^3-2 = 0\]

multiply 1000 both sides, you get :

\[\large 2x^3-2000 = 0\]

Answer 66

now divide 2 both sides, you get :

\[\large x^3-1000 = 0\]

Answer 67

\[\large x^3 = 1000\]
\[\large x = ?\]

Answer 68

1000/3

Answer 69

nopes

Answer 70

i don't know how to undo a cube...

Answer 71

a square is square root right?.. but a cube..

Answer 72

a cube has got a cuberoot too

Answer 73

\[\large x^3 = 1000\]

\[\large x^3 = 10^3\]

\[\large x = ?\]

Answer 74

soo... x=1000

Answer 75

nope
x = 10

Answer 76

i should be able to plug 10 in for x into C' right to get 0?

Answer 77

x=10 is the value that minimizes the average cost per unit

Answer 78

holy cow! That is alot of algebra that i haven't used in, in forever... if at all.. I'm not a mathmatecian for sure!

Answer 79

I really appreciate your patence with me and walking through all the tedious small steps that get me in tears.. i just was so upset tonight with trying to figure this out. I give you infinity medals!

Answer 80

np :) btw, you're good in math as you're asking lot of tough questions
you won't be asking questions if you're not a mathematician haha!

Answer 81

have good sleep :)

Answer 82

lol, i got 4 more questions before bed.. :) thanks for the help!!