Question

um recipiente tem o formato de uma pirâmide triangular regular com lado da base igual a 4 raiz de 3 m. sabendo que sua altura é 3 raiz de 5 m calcule a sua capacidade total.

Answer 1

Calculando a área da base da pirâmide:

$A_{base}=A_{t.~equil\'atero}\\\\\\A_{base}=\dfrac{(a_{base})^{2}\sqrt{3}}{4}\\\\\\A_{base}=\dfrac{(4\sqrt{3})^{2}\sqrt{3}}{4}\\\\\\A_{base}=\dfrac{4^{2}\cdot3\cdot\sqrt{3}}{4}\\\\\\A_{base}=4\cdot3\cdot\sqrt{3}\\\\\\\boxed{\boxed{A_{base}=12\sqrt{3}~m^{2}}}$

Calculando o volume da pirâmide (capacidade total):

$V=\dfrac{1}{3}\cdot A_{base}\cdot h\\\\\\V=\dfrac{1}{3}\cdot12\sqrt{3}\cdot3\sqrt{5}\\\\\\V=\dfrac{1}{1}\cdot4\sqrt{3}\cdot3\sqrt{5}\\\\\\V=3\cdot4\cdot\sqrt{3\cdot5}\\\\\\\boxed{\boxed{V=12\sqrt{15}~m^{3}}}$