Question

PLEASE HELP.
A manufacturer of a brand A jeans has daily production costs of c=0.2x^2-90x+10,700, where C is the total cost (in dollars) and x is the number of jeans produced. How many jeans should be produced each day in order to minimize costs? what is the minimum daily cost?

Answer 1

@aznbobo We want to minimize the cost. This question of maxima and minima. I'll guide you and you'll solve it. You agree with this?

Answer 2

yes

Answer 3

Ok find c'(x)

Answer 4

minimum daily cost would be when you produce nothing...

Answer 5

@aznbobo did you got my point???

Answer 6

@sheg We'll just find out that x=0 won't give us the lowest cost

Answer 7

@aznbobo Did you find c'(x)

Answer 8

i do not know how to do that. >.>

Answer 9

@ash2326 plug in x=0 and see what you get it is fixed cost that incur to the company when the company produces virtually nothing

Answer 10

when i plug in 0 for x i get... 10700

Answer 11

@aznbobo that's not right. Do you know differentiation?

Answer 12

now if you are talking of finding out the minimum cost by finding the total number of jeans produced so in this case we can use the Khun-Tucker condition which states that if the first derivative of any equation is equal to zero and the second derivative is positive then we can get the minimum cost @ash2326

Answer 13

so differentiate the given equation

Answer 14

@aznbobo Do you know parabolas?

Answer 15

Kuhn-Tucker Condition*

Answer 16

yes parabolas

Answer 17

@aznbobo you just need to differentitate the given equation with respect to x

Answer 18

@aznbobo how's the graph of
\[c(x)=0.2x^2-90x+10700\]
Does it have concavity upwards or downwards?

Answer 19

upwards.

Answer 20

on diff. you will get
0.4x - 90 = 0
again diff. wrt x
0.4 that is a positive quantity
so the above equation fulfills the Kuhn-Tucker Condition
so above equation i.e., 0.4x - 90 = 0
0.4x = 90
x = 225
so number of jeans produced would be 225
and by plugging in x = 225 you can calculte the minimum cost

Answer 21

Good you know this function is our cost function. The cost will be minimum at the vertex, can you find the vertex of this graph?

Answer 22

umm i think (225,275)?

Answer 23

How did you find it? Can you explain a little?

Answer 24

i graphed in on the calculator..

Answer 25

You know standard form of a parabola is
\[y=a(x-h)^2+k^2\]
Where h,k is the vertex

here
\[c(x)=a(x-h)^2+k^2\]
We need to convert the equation in this form and we'll get the minimum cost as k and no. of jeans as h. Do you understand this?

Answer 26

so k is already given at 10700, right? h would be... 90?

Answer 27

No it's not easy, we need to convert in that form
We have
\[c(x)=0.2x^2-90x+10700\]
We need it an a form
\[c(x)=\underline{a(x-h)^2}+k^2\]
Notice the underlined term here we need to have a perfect square term (x-h)^2
do you understand the difference?

Answer 28

yes

Answer 29

Great, ok let's get to work now

Answer 30

We have
\[c(x)=0.2x^2-90x+10700\]
now this can be written as
\[c(x)=0.2(x^2-450x)+10700\]
now we have
\[x^2-450\]
We need to convert this to a square term,
Can you do that?

Answer 31

x=\[15\sqrt{2} \] & \[-15\sqrt{2} \]?

Answer 32

@aznbobo We need to convert this to a perfect square term
\[x^2-450x \longrightarrow(x-a)^2\]
Can you do it? If you can't I'll help you:)

Answer 33

mmm i can you help me please? :]

Answer 34

Ok
\[x^2-450x\]
Whenever you need to convert into a perfect square, add and subtract the square of half the coefficient of x
here coefficient of x is 450, so half of it is 225, We'll add and subtract the square of this, so we have
\[x^2-450x+225^2-225^2\]
so We get
\[(x-225)^2-225^2\]
so We'll substitute this for x^2-450x
\[x^2-450x\longrightarrow (x-225)^2-225^2\]
Do you understand this?

Answer 35

okay

Answer 36

Now we have
\[c(x)=0.2((x-225)^2-225^2)+10700\]
It can be written as
\[c(x)=0.2(x-225)^2-0.2 \times 225^2+10700\]
\[c(x)=0.2(x-225)^2-0.2 \times 50625+10700\]
\[c(x)=0.2(x-225)^2- 10125+10700\]
Finally we get
\[c(x)=0.2(x-225)^2+ 575\]
So we have the form of
\[c(x)=a(x-h)^2+k\]
here k is the minimum cost and h is the no. of jeans for minimum cost, the vertex is h,k
here it's h=225 and k=575
so
no. of jeans=225
minimum cost= 575

Answer 37

@aznbobo Do you understand this?

Answer 38

@ash2326 thats what i calculated using the Kuhn_Tucker Condition