De todos os cones que contém uma esfera de raio R, qual tem o menor volume?.... Pergunta de ideia deGregoryKaique

January 2020 1 63 Report

De todos os cones que contém uma esfera de raio R, qual tem o menor volume?

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Comentários (2) Boa tarde Gregory!

Solução!

$\boxed{V= \dfrac{ \pi .r^{2}.h }{3}}\\\\\ Como ~~o~~ cone~~ pode~~ ser~~ decomposto~~ em ~~dois~~ tri\^agulos~~ retangulos. \\\\\\ Geratriz=L\\\\\ Hipotenusa=(h-r)\\\\\\R=Eesfera\\\\\\\\ Base=r\\\\\ (h-r)^{2}=(l)^{2}+(r)^{2}\\\\\\ h^{2}-2hr+r^{2}=l^{2}+r^{2}\\\\\\ h^{2}-2hr+r^{2}-r^{2} =l^{2}\\\\\\ h^{2}-2hr =l^{2}\\\\\\ \boxed{\sqrt{h^{2}-2hr}=l}$

$Por~~semelhanca!\\\\\\



 \dfrac{r}{h}= \dfrac{R}{\sqrt{h^{2}-2hr}} \\\\\\
 r= \dfrac{Rh}{\sqrt{h^{2}-2hr}} \\\\\\


$

Retomando a formula do volume do cone,vamos substituir o valor do raio.

$V= \dfrac{ \pi .r^{2}.h }{3}}\\\\\\\\ V= \dfrac{ \pi .\bigg(\dfrac{Rh}{\sqrt{h^{2}-2hr}}\bigg)^{2} .h }{3}}\\\\\\\\V= \dfrac{ \pi .\bigg(\dfrac{R^{2}h^{2} }{h^{2}-2hr}\bigg) .h }{3}}\\\\\\\\
V=\bigg(\dfrac{\pi R^{2}h^{3} }{3h^{2} -6hr}\bigg)}\\\\\\\\\
 V=\bigg(\dfrac{h(\pi R^{2}h^{2} )}{h(3h-6r)}\bigg) }$

$\boxed{V=\bigg(\dfrac{\pi R^{2}h^{2}}{3h-6r}\bigg) }$

Boa tarde!
Bons estudos!

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