Optimization problem/finding domain and proof

[miglo]

Homework Statement

a metal box with square base a no top holds 1000 cubic centimeters. it is formed by folding up the sides of the flattened pattern picture and seaming up the four sides. the material for the box costs $1.00 per square meter and the cost to seam the sides is 5 cents per meter. Find the dimensions of the box that costs the least to produce.

The Attempt at a Solution

so for this problem I've managed to find the function the gives the cost to make the box
after converting everything to the same units my function is $$C(x)=x^2+\frac{.004}{x}+\frac{.0002}{x^2}$$
so the derivative for the cost function is $$C'(x)=2x-\frac{.004}{x^2}-\frac{.0002}{x^3}$$
setting this equal to zero and using maple, i get $x=0.1494530180$ plus 3 more zeros but one is negative and the other two are complex so in the context of this problem those 3 make no sense
after plotting my function i can see the critical point i have found is in fact the minimum, but I am also asked to find the domain of my cost function and write up a proof that i have found a minimum and this is where I am stuck.

Homework Equations

if the cost function was given to me as just a regular function, one not in the context of this problem then i would say my domain would be all real except x=0, but I don't think it'll work for this

for my proof, i think ill have to use intermediate value theorem and extreme value theorem? but the extreme value theorem only works for closed intervals, and i still haven't found the domain

 

[micromass]

Hi miglo!

I seem to find the function

$$f(x)=x^2+\frac{0.004}{x}+\frac{0.02}{x^2}$$

Certain that you didn't make any mistakes?

Also, your derivative is incorrect...

 

[Ray Vickson]

Homework Statement

a metal box with square base a no top holds 1000 cubic centimeters. it is formed by folding up the sides of the flattened pattern picture and seaming up the four sides. the material for the box costs $1.00 per square meter and the cost to seam the sides is 5 cents per meter. Find the dimensions of the box that costs the least to produce.

The Attempt at a Solution

so for this problem I've managed to find the function the gives the cost to make the box
after converting everything to the same units my function is $$C(x)=x^2+\frac{.004}{x}+\frac{.0002}{x^2}$$
so the derivative for the cost function is $$C'(x)=2x-\frac{.004}{x^2}-\frac{.0002}{x^3}$$
setting this equal to zero and using maple, i get $x=0.1494530180$ plus 3 more zeros but one is negative and the other two are complex so in the context of this problem those 3 make no sense
after plotting my function i can see the critical point i have found is in fact the minimum, but I am also asked to find the domain of my cost function and write up a proof that i have found a minimum and this is where I am stuck.

Homework Equations

if the cost function was given to me as just a regular function, one not in the context of this problem then i would say my domain would be all real except x=0, but I don't think it'll work for this

for my proof, i think ill have to use intermediate value theorem and extreme value theorem? but the extreme value theorem only works for closed intervals, and i still haven't found the domain

Homework Statement

Homework Equations

The Attempt at a Solution

I don't know your units, but when I use b = base of box and h = height of box = 1000/b (h, b in cm) and express all costs in $ I get (using Maple 9.5):
$$\displaystyle C(b) = \frac{2}{5} + \frac{b^2}{10000} + \frac{2}{b} + \frac{400}{b^2}$$

Here, the material cost is 10^(-4) ($/cm^2) and the seaming cost is 5*10^(-4) $/cm.

Putting b = 100*x gives the cost
$$\displaystyle C(x) = x^2 + 0.4 + \frac{0.2}{x} + \frac{.04}{x^2}$$
The derivative C'(x) has the form N(x)/(50*x^3), where N(x) = 100*x^4-x-4. The function N(x) is strictly increasing if x > (1/400)^(1/3), and N is negative at that point. Thus, there is exactly one positive root of N(x) = 0.

RGV

 

[miglo]

well the problem states that the metal box can only hold up to 1000 cm^3, therefore the volume should be 1000 cm^3
but the cost for the material are all stated in meters and not centimeters, so i converted from centimeters to meters

100cm=1m
(100cm)^3=(1m)^3
1,000,000cm^3=1m^3
(1m^3/1,000,000cm^3)*(1000cm^3/1)=1000m^3/1,000,000=0.001m^3

therefore the volume of the metal box in cubic meters is 0.001

but i also know the volume of a box is length*width*height, and in this problem i have length and width being equal since my base is a square, so naming the length and width of my sides x and my height y, i get V=x^2y
but V=0.001=x^2y, so x^2y=0.001 solving for y i get y=0.001/x^2
so my cost function, C(x)=x^2+4x(0.001/x^2)+4*0.05(0.001/x^2)
so C(x)= x^2+0.004/x+0.0002/x^2

did i do something wrong here to get my cost function?

and yeah i just realized my derivative is wrong, it should be C'(x)=2x-0.004/x^2-0.0004/x^3
however this is pretty much the easy part for me, what I am having trouble with is finding the domain of C(x), and proving that i have found a minimum

 

[micromass]

well the problem states that the metal box can only hold up to 1000 cm^3, therefore the volume should be 1000 cm^3
but the cost for the material are all stated in meters and not centimeters, so i converted from centimeters to meters

100cm=1m
(100cm)^3=(1m)^3
1,000,000cm^3=1m^3
(1m^3/1,000,000cm^3)*(1000cm^3/1)=1000m^3/1,000,000=0.001m^3

therefore the volume of the metal box in cubic meters is 0.001

but i also know the volume of a box is length*width*height, and in this problem i have length and width being equal since my base is a square, so naming the length and width of my sides x and my height y, i get V=x^2y
but V=0.001=x^2y, so x^2y=0.001 solving for y i get y=0.001/x^2
so my cost function, C(x)=x^2+4x(0.001/x^2)+4*0.05(0.001/x^2)
so C(x)= x^2+0.004/x+0.0002/x^2

did i do something wrong here to get my cost function?

Sorry, I totally misread the question. I thought it was 5$ a meter instead of 5 cents a meter.

and yeah i just realized my derivative is wrong, it should be C'(x)=2x-0.004/x^2-0.0004/x^3

however this is pretty much the easy part for me, what I am having trouble with is finding the domain of C(x),
[/QUOTE]

Well, in what points in C not defined? Can I evaluate C(2), C(3.02)? (yes). Can I evaluate C(x) for all x? (No)

and proving that i have found a minimum

Prove that your function is decreasing before the "minimum" and increasing after. So prove that your derivative is negative before the "minimum" and positive after.

Or you can calculate the second derivative C''(x) and prove that it is >0 at the "minimum".

 

[Ray Vickson]

well the problem states that the metal box can only hold up to 1000 cm^3, therefore the volume should be 1000 cm^3
but the cost for the material are all stated in meters and not centimeters, so i converted from centimeters to meters

100cm=1m
(100cm)^3=(1m)^3
1,000,000cm^3=1m^3
(1m^3/1,000,000cm^3)*(1000cm^3/1)=1000m^3/1,000,000=0.001m^3

therefore the volume of the metal box in cubic meters is 0.001

but i also know the volume of a box is length*width*height, and in this problem i have length and width being equal since my base is a square, so naming the length and width of my sides x and my height y, i get V=x^2y
but V=0.001=x^2y, so x^2y=0.001 solving for y i get y=0.001/x^2
so my cost function, C(x)=x^2+4x(0.001/x^2)+4*0.05(0.001/x^2)
so C(x)= x^2+0.004/x+0.0002/x^2

did i do something wrong here to get my cost function?

and yeah i just realized my derivative is wrong, it should be C'(x)=2x-0.004/x^2-0.0004/x^3
however this is pretty much the easy part for me, what I am having trouble with is finding the domain of C(x), and proving that i have found a minimum

Your C(x) is wrong. My previous post was in error, and I have corrected it below. The area of the material (from which you cut out 4 corners of size y) is length * width = (x + 2y)^2 = x^2 + 4*y^2 + 4*x*y. You have y = 0.001/x^2, so the cost is C(x) = x^2 + 4*(0.001/x^2)^2 + 4*x*0.001/x^2 + 4*(.05)*0.001/x^2. This will have terms in x^2, 1/x, 1/x^2 and 1/x^4. Therefore, when you put it over a common denominator, the derivative of C(x) will have a polynomial of degree 6 in the numerator, so there will generally be 6 roots. However, plotting shows there is only one real, positive root.

RGV

 

[miglo]

@micromass
well then my domain will be all reals except for x=0 right?

and for the proof could i just use the intermediate value theorem on C'(x) in the interval [0.1,0.2] since C'(0.1)<0 and C'(0.2)>0 then there exists a zero in the interval, namely the x value i found earlier, then since C'(x) goes from negative to positive, that means C(x) decreases then increases after

 

[micromass]

@micromass
well then my domain will be all reals except for x=0 right?

Right!

and for the proof could i just use the intermediate value theorem on C'(x) in the interval [0.1,0.2] since C'(0.1)<0 and C'(0.2)>0 then there exists a zero in the interval, namely the x value i found earlier, then since C'(x) goes from negative to positive, that means C(x) decreases then increases after

sounds good!