Uma formiga esperta está sobre um vértice do cubo e pretende ir até o vértice diagonalmente oposto percorrendo o menor caminho possível sobre a superfície do cubo. Utilizando o applet acima, tente visualizar qual seria esse menor caminho possível na planificação do cubo. Calcule o comprimento desse menor caminho possível em função de r: raio do circulo circunscrito à base do cubo. Esse menor caminho é único?
[b]Visualização na Planificação do Cubo[/b][br]Quando o cubo é planificado, o menor caminho entre dois vértices opostos pode ser visto como uma [b]linha reta[/b] sobre a superfície da planificação. Em vez de percorrer várias faces separadamente, a formiga pode seguir um trajeto direto ao longo da planificação.[br][br][b]Cálculo do Comprimento do Caminho[/b][br]Se [b]r [/b] é o raio do círculo circunscrito à base do cubo, então o lado do quadrado base do cubo é [b]√2 * r[/b]. O menor caminho possível sobre a superfície do cubo pode ser encontrado considerando a planificação em formato de cruz, onde a formiga percorre duas faces consecutivas.[br]Usando o [b]Teorema de Pitágoras[/b], o comprimento do menor caminho será:[br][br][b][img 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[/img][/b][br][b]O Caminho é Único?[/b][br]Não! Existem [b]duas opções equivalentes [/b]para o menor caminho, dependendo da direção escolhida na planificação. A formiga pode seguir por **duas faces consecutivas diferentes**, mas ambas terão o mesmo comprimento [b]√10 * r[/b].