Volume do cone
Definição:
[size=100]De forma resumida, o volume do cone é a medida de capacidade do cone. Assim, o volume dessa figura geométrica é obtido através do produto entre a área da base e a medida da altura, dividindo o resultado por três.[br][size=150][br][b]Considere a situação a seguir:[/b][/size][br][br][justify]Um doce muito famoso e tradicional no Brasil é o canudinho de doce de leite. Ele consiste em uma massa fina frita em formato que lembra um cone que é recheado com doce de leite cremoso. Exatamente por ser muito famoso, o dono de uma confeitaria decidiu produzir e vender esse doce. Para determinar quanto de doce de leite seria necessário fazer, é preciso responder duas perguntas: quantas unidades de canudinhos ele pretende produzir por dia e qual a quantidade de doce de leite necessária para preencher cada canudo. [/justify][/size]
[size=100]Mas como calcular essa quantidade?[/size]
[justify][size=100]Assim como foi feito para determinar o volume de uma pirâmide, no Capítulo anterior, aplicando o princípio de Cavalieri, podemos utilizar o mesmo raciocínio para determinar o volume de um cone. Considere um cone C e uma pirâmide P de mesma altura de medidas [math]h[/math] e bases de mesma área ([math]A_B[/math]-área de base maior), contidas em um plano horizontal [math]\alpha[/math]. Qualquer plano[math]\beta[/math], paralelo ao plano [math]\alpha[/math], distante[math]h'[/math] do vértice e secante aos sólidos C e P determina duas secções transversais de áreas ([math]A_{b1}[/math] e [math]A_{b2}[/math]-área da base menor), respectivamente.[br][br][br]Sabemos que para pirâmides vale a igualdade [math]\frac{A_{b\left(Pirâmide\right)}}{A_B_{\left(Pirâmide\right)}}=\left(\frac{h}{H}\right)^2[/math], prova-se que a relação análoga vale também para cones, ou seja, [math]\frac{A_{b\left(Cone\right)}}{A_{B\left(Cone\right)}}=\left(\frac{h}{H}\right)^2[/math][/size][/justify]Logo, [math]\frac{A_{b\left(Cone\right)}}{A_{B\left(cone\right)}}=\frac{A_{b\left(Pirâmide\right)}}{A_B\left(_{Pirâmide}\right)}\Longrightarrow A_{b\left(Cone\right)}=A_{b\left(Pirâmide\right)}[/math][br]
Assim, pelo princípio de Cavalieri, podemos concluir que o volume da pirâmide P é igual ao volume do cone C e podemos escrever:[br][br] [size=100][math]V_{pirâmide}=V_{cone}=V=\frac{AreadaBase.altura}{3}[/math][/size][br][br][math]V=\frac{1}{3}.A_b.h\Longrightarrow V=\frac{1}{3}.\pi.r^2.h[/math]
Veja um exemplo prático.
Confira a explicação com o Prefº. Murakami.
Exercícios Resolvidos
[justify]Voltando ao problema do recheio de doce de leite, se o dono da doceria fez [math]800ml[/math] de doce de leite, aproximadamente quantos canudos com 3 cm de diâmetro por 8 cm de altura ele poderá preencher? Saiba que [math]1cm^3=1ml.[/math][/justify]
[b]1º Passo: [/b]Identificar cada valor referente a fórmula do volume do cone.[br][br][b]Temos: [/b][br][br]A altura (h)= 8 cm[br]Diâmetro= 3, para encontrar o raio dividimos por 2, então: [math]r=\frac{d}{2}\Longrightarrow r=\frac{3}{2}\Longrightarrow r=1,5[/math][br] [math]\pi\cong3,14[/math][br][br][b]2º[/b] [b]Passo:[/b] aplicar a fórmula : [math]V=\frac{1}{3}.\pi.r^2.h[/math] , então; [math]V=\frac{1}{3}\times3,14\times\left(1,5\right)^2\times8[/math][br][br][math]V=\frac{1}{3}\times3,14\times2,25\times8\Longrightarrow V=\frac{1}{3}\times56,52\Longrightarrow V=\frac{56,52}{3}\Longrightarrow V=18,84m^3[/math][br][br][b]3º Passo: [/b]No 2º passo, achamos o valor referente ao volume de cada canudo de cone, agora basta dividir as 800 ml que o dono da doceria fez por esse resultado:[br][br][math]800\div18,84=42,4628[/math] Ou seja, para as 800 ml de doce de leite que o dono da doceria fez, será necessário aproximadamente 42 Canudos de cone.[br]
[justify]Um fabricante resolveu fazer a embalagem para um de seus produtos no formato de um cone reto, com 8 cm de diâmetro e 12 cm de altura. Qual será a quantidade mínima do material utilizado para cobrir toda a [br]superfície dessa embalagem? Use [math]\pi=3,14[/math]e [math]\sqrt{10}=3,16[/math].[/justify][br][img]data:image/png;base64,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[/img][br]Aplicando o teorema de Pitágoras, temos: [math]g^2=12^2+4^2=144+16=160[/math][br][math]g=\sqrt{160}=4\sqrt{10}\Longrightarrow g=4\sqrt{10}cm[/math][br][br]Vamos agora determinar a área da base [math]\left(A_b\right)[/math]: [math]A_b=\pi\times r^2=\pi\times4^2\Longrightarrow16\pi\Longrightarrow A_b=16\pi cm^2[/math][br][br]Cálculo da área lateral [math](Al)[/math]: [math]Al=\pi\times r\times g=\pi\times4\times4\sqrt{10}=16\pi\sqrt{10}\Longrightarrow Al=16\pi\sqrt{10}cm^2[/math][br][br]Cálculo da área total (St): [math]At=A_b+Al=16\pi+16\pi\sqrt{10}=16\pi(1+\sqrt{10})\Longrightarrow At=16\pi(1+\sqrt{10})cm^2[/math][br][br]Como [math]\sqrt{10}=3,16[/math], obtemos: [math]At=16\times(3,14)\times(1+3,16)=50,24\times(4,16)\cong209[/math][br]
Exercício
1-Em um cone reto, a área da base é [math]9\pi[/math] cm2 e a geratriz mede [math]3\sqrt{10}[/math] cm. Determine o volume do cone.
2-Um cone possui altura medindo 5 cm e diâmetro da base igual a 8 cm. Considerando π = 3, o volume desse cone é igual a:
3-. Um cone circular reto tem 3 cm de raio e 15p cm2 de área lateral. Calcule seu volume.
4-(IDCAP) Um cone que tem o volume V = 37,68 cm³ e cujo raio da base é r = 3 cm. [br][br]Considerando π = 3,14, a medida de g é:[br][br]
5-Um recipiente no formato de um cone possui altura igual a 12 cm e o comprimento da circunferência da base igual a 52,7 cm. Utilizando π = 3,1, o volume desse recipiente, aproximadamente, é de: