OBMEP - Nível 3

2016 - Questão 11
Os quadrados da figura têm lados paralelos e o mesmo centro. O quadrado maior tem lado 10 e o menor tem lado x. Qual é o gráfico que expressa a área da região cinza em função de x?
Opções:
Antes de tudo, movimente o valor de x para compreender melhor a situação expressa na questão.
Vamos entender a composição da figura que pretendemos calcular a área em função de x.
Como é a figura que queremos determinar a área?
É possível perceber que a figura cinza é igual ao triângulo 1 em que tiramos o pedaço relativo ao triângulo 2. Com isso, basta que a gente determine a medida da área usando essas informações em função de x. [br][br]Já sabemos que a área do triângulo 2 é a metade da área do quadrado de lado medindo x.[br]Agora, só falta determinar a altura do triângulo 1, uma vez que já sabemos a medida de sua base.
Como determinar a altura do triângulo 1?
Em outras palavras, podemos dizer que a medida da altura do triângulo 1 será igual à 5 mais a metade de x.
Como continuar?
Existem várias formas diferentes de resolver esta situação. Uma delas é encontrando uma expressão algébrica que represente a área cinza [b]em função[/b] de x, ou seja, em função da medida do lado do quadrado menor. Isso significa que iremos encontrar uma função que associa uma medida de área para cada valor de x entre 0 e 10.
Uma função do 2º grau!
Maravilha! Já sabemos que o gráfico será uma parábola. Como o coeficiente [i]a[/i] é negativo, sabemos que a concavidade desta parábola será para baixo. [br][br]Temos, agora, apenas duas opções[br][center][img]data:image/png;base64,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[/img][/center]Outra informação que podemos considerar é a posição do vértice. [br]No item (b), o vértice não está sobre o eixo e no item (d), está.[br][br]Como a abscissa do vértice é dada em função do coeficiente [i]b[/i] e [i]b[/i][math]\ne[/math]0, podemos afirmar, com certeza, que o vértice não está sobre o eixo y.[br][br]Portanto, por eliminação, nos resta a opção (b) como resposta.
Outra forma de fazer:
Uma outra forma de fazer seria testando valores. [br]Para isso, vamos utilizar uma tabela em que iremos variando os valores de x e verificando as medidas das áreas do triângulo 1, do triângulo 2 e da região cinza.
Testando valores: varie o valor de x para preencher a tabela a seguir.
Clique nas células azul e complete com as medidas das áreas encontradas acima.
Vamos ver se nosso raciocínio está correto. Clique em "Exibir função".
Legal!
Isso significa que nosso raciocínio está correto![br][br]O gabarito, portanto, só pode ser a letra B.[br][br]Muito bem! Se você quiser acessar toda a prova da 1ª fase do nível 3 da OBMEP de 2016 de onde esta questão foi retirada, [url=https://drive.google.com/file/d/11wxnOCAbxYS1d_FAYKAxfcE5-nJ5iZPk/view]clique aqui[/url].[br][br]Para acessar outras provas, acesse a [url=http://www.obmep.org.br/provas.htm]página oficial da OBMEP[/url]. Lá você consegue acessar o gabarito, questões resolvidas e, inclusive, encontrar vídeos com as resoluções de algumas questões.

Information: OBMEP - Nível 3