Optimization, Minima, Open Top Box

[rabbitstorm]

Optimization, Minima, new question:Sheet Alluminum

Homework Statement

A box with a square base and no top must haave a volume of 10000 cm^3. If the smallest dimension in any direction is 5 cm, then determine the dimensions of the box that minimize the amount of material used.

Homework Equations

Volume: x^2y
Surface Area: x^2+4xy

The Attempt at a Solution

Let x represent the length and width of the box
Let y represent height

x>5

I drew a neat little drawing of the box and labeled it according to the statements above.

V=x^2y
10000=x^2y
y=10000/x^2


SA=x^2+4xy
=x^2+4x(10000/x^2)

Now I think that I need to set the SA equation to zero then differentiate but I can't quite remember what I'm doing with it. I'd appreciate anyhelp that could be offered.


~Thanks!

 

[2ltben]

Don't forget to subtract the area of the top from the surface area equation.

 

[Gib Z]

He never added it lol.

Your on the right track. $$SA=x^2+\frac{40000}{x}$$

Differentiate with respect to x and set equal to zero. you get $$2x=\frac{40000}{x^2}$$ Take the x^2 over and we get 2x^3=40000, x^3=20000. Take the cube root, we get around 27.144cm. Sub that value Into the 10000=x^2 y to get y.

 

[rabbitstorm]

wow.. Thanks that make sense I think, been a couple months since we did this in class.

Thanks
~RS

So then x=27.144 cm and y=6.786 cm.
Don't I also sub 5 from x>5 into the equation to tie up loose ends and make sure that it really is 27 cm? oops wrong equation y=13 and change.

 

[rabbitstorm]

A cylindrical can is to hold 500 cm^3 of apple juice the design must take into account that the height must be between 6 and 15 cm, inclusive. How should the can be constructed so that a minimum amount of material will be used in the construction? assume no waste.

V is volume and SA is surface area

so far this is what I have:

15>h>6

v=(pi)r^2h
500=(pi)r^2h
500/(pi)r^2=h

SA=2(pi)r(500/(pi)r^2) + 2(pi)r^2

Now I believe I need to find the derivative and set it to zero then solve for r.

The problem is that I can't quite figure out how I'm supposed to find the derivative if someone could give me a hand understanding the process as I have forgotten and my teacher is busy with the latest stuff.

 

[HallsofIvy]

No, your requirement is that x and y must both be greater than 5 which is true.

 

[rabbitstorm]

No, your requirement is that x and y must both be greater than 5 which is true.

Sorry I forgot to say that I finally got that question finished and it seems to be correct, I had been struggling with it for the last few days and my teacher wasn't able to explain it in a way that helped. Thanks to Gib Z it clicked and I figured it out as far as boxes go, but when I see derivatives and pi I get thrown off.

Thanks,