Question

A área lateral de um cone, quando planificada, é um setor circular de 120º e área 3π cm 2 . Determine o volume e a área total do cone.

Answer 1

Volume do cone:

$\large \text { V = \dfrac{2\pi \sqrt{2} }{3} ~~cm ^3 }$

Área total do cone:

$\large A_t =4 \pi ~cm^2$

                                      $\LargeS\acute{o}lidos ~Geom\acute{e}tricos$

A área do setor circular = S =  3π cm²

O valor de Ф = 120⁰

Encontrar o valor do geratriz  do setor circular.

$\large S = \dfrac{\theta}{360} ~. \pi ~. ~g^2 \\\\\\\large 3\pi = \dfrac{120~. \pi ~. ~g^2}{360} \\\\\\\large 3\pi = \dfrac{1~. \pi ~. ~g^2}{3} \\\\\\\large 3\pi = \dfrac{ \pi ~. ~g^2}{3} \\\\\\\large \dfrac{ \pi ~. ~g^2}{3} = 3\pi \\\\\\\large g^2 \pi = 3\pi ~. ~3 \\\\\\\large g^2 \pi = 9\pi \\\\\\\large g^2 = \dfrac{9\pi }{\pi }$

$\large g^2= 9 \\\\\large g =\sqrt{9} \\\\\large g = 3 ~cm$

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Formula da Área lateral do cone pela geratriz, encontrar o valor do raio.

$\large S_l = \pi ~. r ~. ~g \\\\\\\large 3\pi = \pi ~. r ~. ~3 \\\\\\\large \pi ~. r ~. ~g = 3\pi \\\\\\\large \pi ~. r ~. ~3 = \dfrac{3\pi }{3\pi } \\\\\\\large r = 1 ~cm$

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Encontrar a altura ( h )  do cone por Pitágoras:

$\large h^2 = g^2 + r^2\\\\\large h^2 = 3^2 - 1^2\\\\\large h^2 = 9 - 1\\\\\large h^2 =8\\\\\large h =\sqrt{8} \\\\\large h =2\sqrt{2} ~cm$

Encontrar  o volume do cone:

$\large V = \dfrac{\pi ~. ~r^2 ~. ~h}{3} \\\\\\\large \text { V = \dfrac{\pi ~. ~1^2 ~. ~2\sqrt{2} }{3} }\\\\\\\large \text { V = \dfrac{\pi ~. ~1 ~. ~2\sqrt{2} }{3} }\\\\\\\large \text { V = \dfrac{2\pi \sqrt{2} }{3} ~~cm ^3 }$

Encontrar a área total do cone:

$\large A_t= \pi~. ~r (r + g) \\\\\\\large A_t= \pi~. ~ 1 (1 + 3) \\\\\\\large A_t= \pi~. ~ 1 ~. ~(4) \\\\\\\large A_t =4 \pi~cm^2$

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