Lagrange multiplier problem Watch

a nice man
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Hi there,

I've been working my way through problems on Lagrange multipliers and I'm confused by this one that I've attached.

A rectangular parallelepiped has the surface area formula Surface area = 2(wl + hl + hw)

which I figured would translate in coordinate terms to Surface area = 2(xz + yz +xy).

However, in this problem the surface area of the rectangular parallelepiped is given as Surface area = 8(xz + yz +xy).

Why is this?

Thanks in advance for any help
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ghostwalker
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(Original post by a nice man)
However, in this problem the surface area of the rectangular parallelepiped is given as Surface area = 8(xz + yz +xy).

Why is this?
The parallelepiped is defined by a single point in the x,y,z > 0 octant.

So, what are the lengths of the sides? A sketch may prove useful - a view from above would do, to bring out the important point.

Spoiler:
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2x, 2y, 2z
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a nice man
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(Original post by ghostwalker)
The parallelepiped is defined by a single point in the x,y,z > 0 octant.

So, what are the lengths of the sides? A sketch may prove useful - a view from above would do, to bring out the important point.

Spoiler:
Show


2x, 2y, 2z
I'm not sure if I completely follow you, but I've attached a sketch of a rectangle (for simplicity) that is my attempt at explaining the textbook's answer and interpreting what I think you are suggesting.

Obviously, I haven't got a third-dimension in my sketch but the same principle would hold for a rectangular parallelepiped in three-dimensions.

Is my sketch correct, or have I got the wrong end of the stick?
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ghostwalker
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(Original post by a nice man)
Is my sketch correct, or have I got the wrong end of the stick?
That's correct. The lengths of the sides are 2x, 2y, and 2z.

Hence the surface area is....
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dhrubajit
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Dear Student,
You took the origin (0,0,0) as a corner and side lengths x,y,z.
But in the problem, parallelepiped is symmetrical about x, y, z plane.
each side is of length 2x, 2y, 2z. x unit in positive side and x unit in negative side. so the origin the the center of the parallelepiped.
So the surface area
S=2[(2x)(2y)+(2y)(2z)+(2z)(2x)]=8(xy+yz+zx)
If you need details step by step solution, contact me in email dhrubajit[at]gmail.com
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a nice man
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Thanks guys. I tried to rep your post too ghost but apparently I've 'rated' you too many times already
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