# Help with Minimum Area

#### karisrou

1) A cardboard box of 32in^3 volume with a square base and open top is to be constructed. Find the minimum area of cardboard needed.

Since it has a square base, length and width must be the same, variable b for base.

So volume could be written as: hb^2=32 and surface area as b^2+4bh=min.

Once I get here, I''m not quite sure what to do.
Any help is greatly appreciated!

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#### Wretchosoft

You want to get your minimum as a function of one variable. Use substitution to make that happen.

#### karisrou

Alright cool. I've got b^2 + 128/b^2 = min

Should I take the derivative now?

#### Dick

Science Advisor
Homework Helper
You might want to make sure that's correct before you take the derivative. You've got h*b^2=32 and you want to minimize b^2+4*b*h. Are you sure that means b^2+128/b^2 should be minimized?

#### karisrou

But aren't I supposed to substitute h into b2 + 4bh? and then solve for the critical point? Or would I just take the derivative of that? And then if thats what I'm supposed to do, do i take the derivative in relation to b or h?

#### Dick

Science Advisor
Homework Helper
But aren't I supposed to substitute h into b2 + 4bh? and then solve for the critical point? Or would I just take the derivative of that? And then if thats what I'm supposed to do, do i take the derivative in relation to b or h?
You are supposed to do exactly that thing. But are you sure substituting h=32/b^2 into b^2+4hb gives you b^2+128/b^2?

#### karisrou

Right, so b2 + 4b(32/b2)

Which simplifies to

b2 + 128b/b2

Which simplifies to

b2 + 128/b

#### Dick

Science Advisor
Homework Helper
Right, so b2 + 4b(32/b2)

Which simplifies to

b2 + 128b/b2

Which simplifies to

b2 + 128/b
Right. Now take the derivative.

#### karisrou

so 2b - 128/b2

and then would you multiply everything by b2 to clear the denominator? giving you 2b3 - 128 = 0
So b = 3rt64

So 4?

Then if b = 4

h must equal 2, per substitution into the original equation.

So then the areas of the sides are 8, and the base is 16, so the total area is 8 x 4 + 16 = 48

Is this right?

#### Dick

Science Advisor
Homework Helper
It looks good to me. But I've made mistakes before. Never hurts to double check. Seem ok to you?

#### karisrou

Yeah, everything looks right. Thanks very much for your help!

#### Mark44

Mentor
Right, so b2 + 4b(32/b2)

Which simplifies to

b2 + 128b/b2

Which simplifies to

b2 + 128/b
It would be a very good idea to work with equations, in part to help keep you focused on what you're doing.

You're trying to minimize the area as a function of b, or A(b) = b2 + 4b(32/b2) = b2 + 128/b

Then when you take the derivative, you get A'(b) = 2b - 128/b2

To find the critical point, you're going to set A'(b) to 0, so 2b - 128/b2 = 0
You can multiply both sides of that equation by b2 to get 2b3 - 128 = 0, and you can then solve that equation for b.

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