Question

MATEMATICA
(obs> Com resoluçao)

1-Calcule a area total e a diagonal de um cubo cujo volume é igual a 216 cm3

2-Calculo o volume de um prisma de base hexagonal regular sabendo que o perimetro de sua base é igual a 42 cm e que sua altura é igual a 10cm

GRATA

Answer 1

Resposta:

Explicação passo-a-passo:

1)

Volume do cubo é dado por:

V = aresta³

V = a³

===

V = a³

216 =  a³

a³ = 216

a = ∛216

a =∛2³.3³  ( Corta os expoente com o indice da raiz )

a = 2.3

a = 6 cm

Área total:

At = 6a²

At = 6.6²

At = 6.36

At = 216 cm²

Diagonal do Cubo:

D = a√3

D = 6√3 cm

===

2)

Encontrar a medida da aresta da base:

a = perimetro  /  6

a = 42 / 6

a = 7 cm

Encontrar a área da base do hexágono:

$A_b = \dfrac{3.a^2\sqrt{3} }{2} \\ \\ \\ A_b = \dfrac{3.7^2\sqrt{3} }{2} \\ \\ \\ A_b = \dfrac{3.49\sqrt{3} }{2} \\ \\ \\ A_b = \dfrac{147\sqrt{3} }{2}~ cm^2$

===

Encontrar o valor do Volume do prisma:

$V = Area ~da ~base ~. altura \\ \\\\ V = A_b ~. ~ h\\ \\ \\ V = ( \dfrac{147\sqrt{3} }{2} )~.~10\\ \\ \\ V = 147\sqrt{3} . 10. \dfrac{1}{2} \\ \\ \\ V = 147\sqrt{3} . \dfrac{10}{2} \\ \\ \\ V = 147\sqrt{3} ~ . ~5\\ \\ \\ V = 735\sqrt{3} ~ cm^3$

Answer 2

$\sf \large{a=\sqrt[\sf 3]{216}}\\\\\\\sf \large{a=\sqrt[\sf 3]{\sf 6^3}}\\\\\\\sf a=6$

Área

$\sf At = 6a^2 \\\\\\At\:=\:6\cdot 6^2\\\\\\ \boxed{\sf S=\{At=216cm^2\}}$

$\sf Diagonal\\\\\\\sf D=\large{\sqrt[\sf a]{\sf 3} }\\\\\\\boxed{\sf S=\{ D=\large{\sqrt[\sf 6]{\sf 3} \}}}$

   

 2.

$\sf Aresta\\\\\\\sf a=\dfrac{42}{6}\\\\\\42\div6\\\\\\\boxed{\sf S=\{ 7\}}$

$\sf base \ do \ hex\acute{a}gono\\\\\\\iff \displaystyle A_b=\frac{3\cdot \:a^2\sqrt{3}}{2}\\\\\\\iff A_b=\frac{3\cdot \:7^2\sqrt{3}}{2}\\\\\\\iff 3\cdot7^2\sqrt{3} =147\sqrt{3} \\\\\\\boxed{\sf S=\{ A_b=\frac{147\sqrt{3}}{2}CM^2\}}$

$\sf Volume \ do \ prisma\\\\\\\sf F\acute{o}rmula >>> \boxed{\sf V=A_b\cdot h}\\\\\\\sf \displaystyle V=\left(\frac{147\sqrt{3}}{2}\right)\cdot \:10\\\\\\\sf V=\frac{147\sqrt{3}}{2}\cdot \:10\\\\\\\sf \iff a\cdot \frac{b}{c}=\frac{a\:\cdot \:b}{c}\\\\\\\sf V=\frac{147\sqrt{3}\cdot \:10}{2}\\\\\\\sf V=\frac{1470\sqrt{3}}{2}\\\\\\\boxed{\sf S=\{ V=735\sqrt{3}~CM^2\}}$