Problema (nível 3): A Alice queria uma toalha rectangular, mas só tinha uma redonda (circular). Então dobrou-a em 4 partes iguais e cortou-a da forma indicada a tracejado na figura. Qual a medida da área do tecido desperdiçado? (image1)


Conteúdo referenciado:
[image1]
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Resolução: Sejam $r$ a medida do raio da circunferência e $[ABC]$ o triângulo rectângulo em $A$ como se indica na figura. Pelo Teorema de Pitágoras, $$r^2=\overline{CB}^2={\overline{CA}}^2+{\overline{AB}}^2= (r-20)^2+40^2.$$ Resolvendo esta equação conclui-se que $r=50$ e $\overline{AC}=30$. Logo a área da toalha rectangular mede $4\times 40\times 30=4800$ $cm^2$ e a área da toalha redonda mede $\pi\times 50^2=2500\pi$ $cm^2$. Portanto a medida da área do tecido desperdiçado é $2500\pi-4800\approx 3054$ $cm^2$. (image2)


Conteúdo referenciado:
[image2]
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