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Sos help with maximum and minimum

leprofece

Member
Jan 23, 2014
241
Between all the cones whose generatrix length given is L, determine the one with the highest volume?
:confused:
the answer is h= L/sqrt(3) and R = Sqrt(6)L/2
 

mente oscura

Well-known member
Nov 29, 2013
172
Re: Sos help whit maximun and minimun

Between all the cones whose generatrix length given is L, determine the one with the highest volume?
:confused:
the answer is h= L/sqrt(3) and R = Sqrt(6)L/2
Hello.

That solution is not me.

[tex]L^2=R^2+h^2[/tex](*)

[tex]Cone \ volume= V =\frac{\pi R^2 h}{3}[/tex]

[tex]V=\dfrac{\pi (L^2-h^2)h}{3}=\dfrac{\pi h L^2-\pi h^3}{3}[/tex]

[tex]\dfrac{d(V)}{d(h)}=\dfrac{\pi L^2-3 \pi h^2}{3}=0[/tex]

[tex]\cancel{\pi} L^2=3 \cancel{\pi} h^2 \rightarrow{}h=\dfrac{L}{\sqrt{3}}[/tex]

[tex]\dfrac{d_2(V)}{d(h)}=- \dfrac{6 \pi h}{3} \rightarrow{} h=\dfrac{L}{\sqrt{3}}= Is \ maximun[/tex]

For (*):

[tex]R^2=L^2-\dfrac{L^2}{3}[/tex]

[tex]R=\dfrac{\sqrt{2}L}{\sqrt{3}}=\dfrac{\sqrt{2} \sqrt{3}L}{\sqrt{3} \sqrt{3}}=\dfrac{\sqrt{6}L}{3}[/tex]

Regards.
 
Last edited:

leprofece

Member
Jan 23, 2014
241
Re: Sos help whit maximun and minimun

hello.

That solution is not me.

[tex]l^2=r^2+h^2[/tex](*)

[tex]cone \ volume= v =\frac{\pi r^2 h}{3}[/tex]

[tex]v=\dfrac{\pi (l^2-h^2)h}{3}=\dfrac{\pi h l^2-\pi h^3}{3}[/tex]

[tex]\dfrac{d(v)}{d(h)}=\dfrac{\pi l^2-3 \pi h^2}{3}=0[/tex]

[tex]\cancel{\pi} l^2=3 \cancel{\pi} h^2 \rightarrow{}h=\dfrac{l}{\sqrt{3}}[/tex]

[tex]\dfrac{d_2(v)}{d(h)}=- \dfrac{6 \pi h}{3} \rightarrow{} h=\dfrac{l}{\sqrt{3}}= is \ maximun[/tex]

for (*):

[tex]r^2=l^2-\dfrac{l^2}{3}[/tex]

[tex]r=\dfrac{\sqrt{2}l}{\sqrt{3}}=\dfrac{\sqrt{2} \sqrt{3}l}{\sqrt{3} \sqrt{3}}=\dfrac{\sqrt{6}l}{3}[/tex]

regards.
thanks a lot