- #1

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Among all square-based pyramids which have volume V = 1, which one has the smallest surface area?

- Thread starter arcnets
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- #1

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Among all square-based pyramids which have volume V = 1, which one has the smallest surface area?

- #2

mathman

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a= length of base side

h= height of pyramid

t= altitude of triangle face

from this t

Then

V (volume) = a

S (surface area) =a

Set V=1, use the above formula for t, and let x=a

S=x+(36/x+x

S'=1+(x-18/x

Set S'=0, we get x

To verify that this is a minimum (not maximum or horizontal inflection), observe that:

x near 0, S approx 6/x

x gets large, S approx 2x.

Finally:

S=(9/2)

I suggest you work this through to understand how it goes.

- #3

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I don't seem to understand this step:

S=x+(36/x+x

S'=1+(x-18/x

Shouldn't it read

S'=1+(x-18/x

...or something?

- #4

mathman

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- #5

Hurkyl

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Er, isn't it right as is?

dividing by z^{1/2} is the same as multiplying by z^{-1/2}

dividing by z

- #6

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Oops! Yes. I overlooked the /.

- #7

mathman

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Hurkl got it right. When I read arcnets comment, I also forgot I had put in the /.

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