Matemática, perguntado por matheusaalmeidax2638, 7 meses atrás

Num cone, de 8 cm de altura, a geratriz é igual a 5/3
do raio da base.
Calcule seu volume.

Soluções para a tarefa

Respondido por Helvio

$\large\text{$O ~volume ~do ~cone ~ \Rightarrow ~ V = 96 ~\pi ~cm^3$}$

$\large\text{$S\acute{o}lidos ~Geom\acute{e}tricos $}$

Encontrar o valor do raio da base do cone

$g^2 = r^2 + h^2\\\\\\(r~. ~\dfrac{5}{3} )^2 = r^2 + 8^2\\\\\\\dfrac{25r^2}{9} = r^2 + 64\\\\\\25r^2 = (r^2 + 64) ~. ~9\\\\\\25r^2 = 9r^2 + 576\\\\\\ 25r^2 - 9r^2 = 576\\\\\\16r^2 = 576\\\\\\ r^2 = \dfrac{576}{16} \\\\\\r^2 = 36\\\\\\r = \sqrt{36} \\\\\\r = 6 ~cm$

Raio da base = 6 cm

Altura ( h ) = 8 cm

$V = \dfrac{1}{3} \pi. r^2. h\\\\V = \dfrac{1}{3} \pi. 6^2. 8\\\\V = \dfrac{1}{3} \pi. 36 . 8\\\\V = \dfrac{1}{3} \pi. 288\\\\V = \dfrac{288}{3} \pi\\\\V = 96 cm³$

$V = \dfrac{1}{3} ~\pi~. ~r^2~. ~h\\\\\\V = \dfrac{1}{3} ~\pi~. ~6^2~. ~8\\\\\\V = \dfrac{1}{3} ~\pi~. ~36~. ~8\\\\\\V = \dfrac{1}{3} ~\pi~. ~288\\\\\\V = \dfrac{288}{3} ~\pi\\\\\\V = 96 ~\pi ~cm^3$