Question

A geratriz de um cone circular reto forma um angulo de 60 graus com a base do cone. Determine a área lateral e o volume desse cone, sabendo que ele tem 12cm de altura.

Answer 1

h² = g² - r²
sen 60º = h/g
√3/2 = 12/g
g√3 = 2.12
g√3 = 24
g = 24√3/3
g = 8√3

tg 60º = h/r
h/r = √3
12/r = √3
r√3 = 12
r = 12√3/3
r = 4√3

AL = π.r.g
AL = π.4√3.8√3
AL = 96π cm²

Answer 2

Calculando o raio do cone:

$cos~60\º=\dfrac{r}{g}\\\\\\\dfrac{1}{2}=\dfrac{r}{8\sqrt{3}}\\\\\\r=\dfrac{8\sqrt{3}}{2}\\\\\\\boxed{\boxed{r=4\sqrt{3}~cm}}$

Calculando a geratriz do cone:

$sen~60\º=\dfrac{h}{r}\\\\\\\dfrac{\sqrt{3}}{2}=\dfrac{12}{r}$

Multiplicando em cruz:

$\sqrt{3}\cdot g=2\cdot12\\\\\\g=\dfrac{24}{\sqrt{3}}\\\\\\g=\dfrac{24\sqrt{3}}{3}\\\\\\\boxed{\boxed{g=8\sqrt{3}~cm}}$
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Calculando a área lateral do cone:

$A_{l}=\pi rg\\A_{l}=\pi\cdot4\sqrt{3}\cdot8\sqrt{3}\\A_{l}=32\pi\cdot3\\A_{l}=96\pi~cm^{2}$

Calculando o volume do cone:

$v=\dfrac{A_{b}\cdot h}{3}\\\\\\v=\dfrac{\pi\cdot(4\sqrt{3})^{2}\cdot12}{3}\\\\\\v=\pi\cdot(\sqrt{48})^{2}\cdot4\\\\\\v=\pi\cdot48\cdot4\\\\\\\boxed{\boxed{v=192\pi~cm^{3}}}$