Maximum volume

L

Loki123

Full Member
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Sep 22, 2021
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790
So I got something wrong here and I don't know what. I have suggested answers for this question, they all include a but not under square root. Where did I go wrong here? IMG_20220504_184920.jpgIMG_20220504_184926.jpgIMG_20220504_184930.jpg
 
I can see at least one typo: in the middle of the second page your last expression for [imath]V[/imath] has incorrect coefficient for the [imath]\;\;x^2\sqrt{x}\;\;[/imath] term
 
blamocur said:
I can see at least one typo writo: in the middle of the second page your last expression for [imath]V[/imath] has incorrect coefficient for the [imath]\;\;x^2\sqrt{x}\;\;[/imath] term
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I plagiarized that term from Cubist and got a copyright.
 
pyramid_max.jpg
[imath]b = (a-x)\sqrt{2}[/imath]

[imath]s^2 = (a^2+x^2) - \dfrac{b^2}{4} = \dfrac{a^2+x^2}{2} - ax[/imath]

[imath]h^2 = s^2 - \left(\dfrac{a-x}{\sqrt{2}}\right)^2 = \dfrac{a^2+x^2}{2} - ax - \dfrac{a^2-2ax+x^2}{2} = ax[/imath]

[imath]V = \dfrac{1}{3}b^2 h = \dfrac{2(a-x)^2}{3} \cdot \sqrt{ax}[/imath]

I get max volume when [imath]x = \dfrac{a}{5}[/imath]
 
skeeter said:
View attachment 32576
[imath]b = (a-x)\sqrt{2}[/imath]

[imath]s^2 = (a^2+x^2) - \dfrac{b^2}{4} = \dfrac{a^2+x^2}{2} - ax[/imath]

[imath]h^2 = s^2 - \left(\dfrac{a-x}{\sqrt{2}}\right)^2 = \dfrac{a^2+x^2}{2} - ax - \dfrac{a^2-2ax+x^2}{2} = ax[/imath]

[imath]V = \dfrac{1}{3}b^2 h = \dfrac{2(a-x)^2}{3} \cdot \sqrt{ax}[/imath]

I get max volume when [imath]x = \dfrac{a}{5}[/imath]
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With my way I keep getting a different H. Why is my way wrong???IMG_20220505_084246.jpg
 
Loki123 said:
With my way I keep getting a different H. Why is my way wrong???View attachment 32580
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I believe your computation of [imath]H[/imath] is correct, but @skeeter's comutatoin of [imath]s^2[/imath] is incorrect, which leads to incorrect value of [imath]H[/imath]. Interestingly, that mistake does not effect the computation of optimal [imath]x[/imath].
 
blamocur said:
I believe your computation of [imath]H[/imath] is correct, but @skeeter's comutatoin of [imath]s^2[/imath] is incorrect, which leads to incorrect value of [imath]H[/imath]. Interestingly, that mistake does not effect the computation of optimal [imath]x[/imath].
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try again (damn signs) ...

[imath](\sqrt{a^2+x^2})^2 - \left(\dfrac{b}{2}\right)^2 = s^2[/imath]

[imath]a^2 + x^2 - \dfrac{[(a-x)\sqrt{2}]^2}{4} = s^2[/imath]

[imath]a^2 + x^2 - \dfrac{a^2 - 2ax + x^2}{2} = s^2[/imath]

[imath]a^2 + x^2 - \left(\dfrac{a^2}{2} - ax + \dfrac{x^2}{2}\right) = s^2[/imath]

[imath]\dfrac{a^2}{2} + \dfrac{x^2}{2} + ax = s^2[/imath]


[imath]h^2 = s^2 - \dfrac{b^2}{4}[/imath]

[imath]h^2 = \dfrac{a^2}{2} + \dfrac{x^2}{2} + ax - \dfrac{a^2-2ax+x^2}{2}[/imath]

[imath]h^2 = 3ax \implies h = \sqrt{3ax}[/imath]

[imath]V = \dfrac{1}{3}b^2 h = \dfrac{2\sqrt{3}}{3}(a-x)^2 \cdot \sqrt{ax}[/imath]

the constant factor won't affect computation of the value of x which yields a max volume
 
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