Problema (nível 3): O Alfredo encaixou 4 dados e 1 berlinde no fundo dum copo, com a seguinte disposição: (image1) Mas agora, para recuperar o seu berlinde preferido o Alfredo tem de partir o copo e para que a sua mãe não descubra decide comprar outro copo igual. Sabendo que o raio do berlinde mede 1 cm, quanto mede o diâmetro da base do copo que o Alfredo tem que comprar?


Conteúdo referenciado:
[image1]
\begin{picture}(0.00,100.00)(-300,30) \multiput(80.00,130.00)(1.60,-0.10){4}{\line(1,0){1.60}} \multiput(86.39,129.59)(0.57,-0.11){11}{\line(1,0){0.57}} \multiput(92.68,128.36)(0.36,-0.12){17}{\line(1,0){0.36}} \multiput(98.76,126.35)(0.24,-0.12){24}{\line(1,0){0.24}} \multiput(104.54,123.57)(0.18,-0.12){30}{\line(1,0){0.18}} \multiput(109.91,120.07)(0.14,-0.12){35}{\line(1,0){0.14}} \multiput(114.78,115.92)(0.12,-0.13){36}{\line(0,-1){0.13}} \multiput(119.09,111.17)(0.12,-0.17){31}{\line(0,-1){0.17}} \multiput(122.76,105.92)(0.12,-0.23){25}{\line(0,-1){0.23}} \multiput(125.72,100.24)(0.12,-0.32){19}{\line(0,-1){0.32}} \multiput(127.93,94.23)(0.12,-0.52){12}{\line(0,-1){0.52}} \multiput(129.36,87.98)(0.10,-1.06){6}{\line(0,-1){1.06}} \multiput(129.97,81.60)(-0.10,-3.20){2}{\line(0,-1){3.20}} \multiput(129.77,75.20)(-0.11,-0.70){9}{\line(0,-1){0.70}} \multiput(128.75,68.87)(-0.11,-0.38){16}{\line(0,-1){0.38}} \multiput(126.92,62.73)(-0.12,-0.27){22}{\line(0,-1){0.27}} \multiput(124.33,56.87)(-0.12,-0.20){28}{\line(0,-1){0.20}} \multiput(121.01,51.39)(-0.12,-0.15){34}{\line(0,-1){0.15}} \multiput(117.01,46.38)(-0.12,-0.12){38}{\line(-1,0){0.12}} \multiput(112.41,41.93)(-0.16,-0.12){32}{\line(-1,0){0.16}} \multiput(107.28,38.10)(-0.21,-0.12){27}{\line(-1,0){0.21}} \multiput(101.69,34.95)(-0.28,-0.11){21}{\line(-1,0){0.28}} \multiput(95.76,32.55)(-0.44,-0.12){14}{\line(-1,0){0.44}} \multiput(89.56,30.92)(-0.91,-0.12){7}{\line(-1,0){0.91}} \put(83.20,30.10){\line(-1,0){6.41}} \multiput(76.80,30.10)(-0.91,0.12){7}{\line(-1,0){0.91}} \multiput(70.44,30.92)(-0.44,0.12){14}{\line(-1,0){0.44}} \multiput(64.24,32.55)(-0.28,0.11){21}{\line(-1,0){0.28}} \multiput(58.31,34.95)(-0.21,0.12){27}{\line(-1,0){0.21}} \multiput(52.72,38.10)(-0.16,0.12){32}{\line(-1,0){0.16}} \multiput(47.59,41.93)(-0.12,0.12){38}{\line(-1,0){0.12}} \multiput(42.99,46.38)(-0.12,0.15){34}{\line(0,1){0.15}} \multiput(38.99,51.39)(-0.12,0.20){28}{\line(0,1){0.20}} \multiput(35.67,56.87)(-0.12,0.27){22}{\line(0,1){0.27}} \multiput(33.08,62.73)(-0.11,0.38){16}{\line(0,1){0.38}} \multiput(31.25,68.87)(-0.11,0.70){9}{\line(0,1){0.70}} \multiput(30.23,75.20)(-0.10,3.20){2}{\line(0,1){3.20}} \multiput(30.03,81.60)(0.10,1.06){6}{\line(0,1){1.06}} \multiput(30.64,87.98)(0.12,0.52){12}{\line(0,1){0.52}} \multiput(32.07,94.23)(0.12,0.32){19}{\line(0,1){0.32}} \multiput(34.28,100.24)(0.12,0.23){25}{\line(0,1){0.23}} \multiput(37.24,105.92)(0.12,0.17){31}{\line(0,1){0.17}} \multiput(40.91,111.17)(0.12,0.13){36}{\line(0,1){0.13}} \multiput(45.22,115.92)(0.14,0.12){35}{\line(1,0){0.14}} \multiput(50.09,120.07)(0.18,0.12){30}{\line(1,0){0.18}} \multiput(55.46,123.57)(0.24,0.12){24}{\line(1,0){0.24}} \multiput(61.24,126.35)(0.36,0.12){17}{\line(1,0){0.36}} \multiput(67.32,128.36)(0.91,0.12){14}{\line(1,0){0.91}} \multiput(33,65)(0,30){2}{\line(1,0){94}} \multiput(33,65)(94,0){2}{\line(0,1){30}} \multiput(65,33)(30,0){2}{\line(0,1){94}} \multiput(65,33)(0,94){2}{\line(1,0){30}} \put(80,80){\circle{30}} \end{picture}

Resolução: Consideremos os pontos $A$, $B$ e $C$ como indicado na figura, onde $C$ é o centro das circunferências e $A$ o ponto de tangência da circunferência interior com um dos quadrados. O triângulo $[CAB]$ é rectângulo em $A$ e aplicando o teorema de Pitágoras vem $\overline{CB}^{2}=\overline{CA}^{2}+\overline{AB}^{2}=1+\overline{AB}^{2}$. Como $\overline{AD}=1$ e $\overline{DB}=2$ então $\overline{AB}=3$ e $\overline{CB}=\sqrt{10}$. Portanto o Alfredo tem que comprar um copo cujo diâmetro da base seja $2\sqrt{10}$ cm. (image2)


Conteúdo referenciado:
[image2]
\begin{picture}(0.00,0.00)(-400,30) \multiput(80.00,130.00)(1.60,-0.10){4}{\line(1,0){1.60}} \multiput(86.39,129.59)(0.57,-0.11){11}{\line(1,0){0.57}} \multiput(92.68,128.36)(0.36,-0.12){17}{\line(1,0){0.36}} \multiput(98.76,126.35)(0.24,-0.12){24}{\line(1,0){0.24}} \multiput(104.54,123.57)(0.18,-0.12){30}{\line(1,0){0.18}} \multiput(109.91,120.07)(0.14,-0.12){35}{\line(1,0){0.14}} \multiput(114.78,115.92)(0.12,-0.13){36}{\line(0,-1){0.13}} \multiput(119.09,111.17)(0.12,-0.17){31}{\line(0,-1){0.17}} \multiput(122.76,105.92)(0.12,-0.23){25}{\line(0,-1){0.23}} \multiput(125.72,100.24)(0.12,-0.32){19}{\line(0,-1){0.32}} \multiput(127.93,94.23)(0.12,-0.52){12}{\line(0,-1){0.52}} \multiput(129.36,87.98)(0.10,-1.06){6}{\line(0,-1){1.06}} \multiput(129.97,81.60)(-0.10,-3.20){2}{\line(0,-1){3.20}} \multiput(129.77,75.20)(-0.11,-0.70){9}{\line(0,-1){0.70}} \multiput(128.75,68.87)(-0.11,-0.38){16}{\line(0,-1){0.38}} \multiput(126.92,62.73)(-0.12,-0.27){22}{\line(0,-1){0.27}} \multiput(124.33,56.87)(-0.12,-0.20){28}{\line(0,-1){0.20}} \multiput(121.01,51.39)(-0.12,-0.15){34}{\line(0,-1){0.15}} \multiput(117.01,46.38)(-0.12,-0.12){38}{\line(-1,0){0.12}} \multiput(112.41,41.93)(-0.16,-0.12){32}{\line(-1,0){0.16}} \multiput(107.28,38.10)(-0.21,-0.12){27}{\line(-1,0){0.21}} \multiput(101.69,34.95)(-0.28,-0.11){21}{\line(-1,0){0.28}} \multiput(95.76,32.55)(-0.44,-0.12){14}{\line(-1,0){0.44}} \multiput(89.56,30.92)(-0.91,-0.12){7}{\line(-1,0){0.91}} \put(83.20,30.10){\line(-1,0){6.41}} \multiput(76.80,30.10)(-0.91,0.12){7}{\line(-1,0){0.91}} \multiput(70.44,30.92)(-0.44,0.12){14}{\line(-1,0){0.44}} \multiput(64.24,32.55)(-0.28,0.11){21}{\line(-1,0){0.28}} \multiput(58.31,34.95)(-0.21,0.12){27}{\line(-1,0){0.21}} \multiput(52.72,38.10)(-0.16,0.12){32}{\line(-1,0){0.16}} \multiput(47.59,41.93)(-0.12,0.12){38}{\line(-1,0){0.12}} \multiput(42.99,46.38)(-0.12,0.15){34}{\line(0,1){0.15}} \multiput(38.99,51.39)(-0.12,0.20){28}{\line(0,1){0.20}} \multiput(35.67,56.87)(-0.12,0.27){22}{\line(0,1){0.27}} \multiput(33.08,62.73)(-0.11,0.38){16}{\line(0,1){0.38}} \multiput(31.25,68.87)(-0.11,0.70){9}{\line(0,1){0.70}} \multiput(30.23,75.20)(-0.10,3.20){2}{\line(0,1){3.20}} \multiput(30.03,81.60)(0.10,1.06){6}{\line(0,1){1.06}} \multiput(30.64,87.98)(0.12,0.52){12}{\line(0,1){0.52}} \multiput(32.07,94.23)(0.12,0.32){19}{\line(0,1){0.32}} \multiput(34.28,100.24)(0.12,0.23){25}{\line(0,1){0.23}} \multiput(37.24,105.92)(0.12,0.17){31}{\line(0,1){0.17}} \multiput(40.91,111.17)(0.12,0.13){36}{\line(0,1){0.13}} \multiput(45.22,115.92)(0.14,0.12){35}{\line(1,0){0.14}} \multiput(50.09,120.07)(0.18,0.12){30}{\line(1,0){0.18}} \multiput(55.46,123.57)(0.24,0.12){24}{\line(1,0){0.24}} \multiput(61.24,126.35)(0.36,0.12){17}{\line(1,0){0.36}} \multiput(67.32,128.36)(0.91,0.12){14}{\line(1,0){0.91}} \multiput(33,65)(0,30){2}{\line(1,0){94}} \multiput(33,65)(94,0){2}{\line(0,1){30}} \multiput(65,33)(30,0){2}{\line(0,1){94}} \multiput(65,33)(0,94){2}{\line(1,0){30}} \put(80,80){\circle{30}} \put(80,80){\line(0,1){15}} \multiput(80,80)(0.3847,0.123){124}{\line(1,0){0.3}} \put(80,77){\makebox(0,0)[rc]{\scriptsize $C$}} \put(83,100){\makebox(0,0)[rc]{\scriptsize $A$}} \put(137,96){\makebox(0,0)[rc]{\scriptsize $B$}} \put(97,100){\makebox(0,0)[lc]{\scriptsize $D$}} \end{picture}