Optimization Part III

H

Hckyplayer8

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Jun 9, 2019
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A box with a square base and no top has a volume of 32ft3. What dimensions use the lest amount of material (give a min surface area)?

V = (L)(W)(H)
A = [2(H)(W)] + [2(H)(L)] + (W)(L)

So subbing x and y values for the above...we know the base is square thus

V = (x2)(y)
A = 4xy + x2

V = 32 so

32 = x2 (y)

y = 32 / x2

S (A) = x2 + 4x(32 / x2) = x2 + 128 / x

S'(A) = 2x - 128 / x2

How am I looking so far?
 
You will get the correct answer if you make no mistakes but I do have one problem with your work/notation. Why S(A) and not S(x), after all YOUR S equals something in terms of x, namely x2 + 4x(32 / x2) = x2 + 128 / x ??
 
Jomo said:
You will get the correct answer if you make no mistakes but I do have one problem with your work/notation. Why S(A) and not S(x), after all YOUR S equals something in terms of x, namely x2 + 4x(32 / x2) = x2 + 128 / x ??
Click to expand...

I was just mimicking an example problem that presented itself in that way. S (A) despite using x and y as the variables.
 
Continuing on.

Setting the derivative to 0 results in x3 = 64

Thus x = 4

Plugging that back into my y value results in 2 which checks out when inserted into the volume equation.
 
Jomo said:
All looks good!
Click to expand...

Thank you for the help Jomo.

The example is posted here.

I did review the video and I was partially wrong. The presenter started out with SA = x2 + 4xy for surface area.

What I didn't catch was the presenter flipped to differentiating based on variable x and featured the correct notation S (x)
 
I knew that it was SA and not S(A). In my opininion if you suffered by having to find the video and then view the video you will remember that SA stands for surface area than if I just told you.
 
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