\n| Custo (\u20ac\/kg)<\/td>\n | 10<\/td>\n | 5<\/td>\n | –<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n As inc\u00f3gnitas<\/strong> do problema s\u00e3o:<\/p>\n\n- x<\/em>,\u00a0a quantidade di\u00e1ria de granulado por animal (em kg)<\/li>\n
- y<\/em>, a quantidade di\u00e1ria de\u00a0farinha por animal (em kg)<\/li>\n<\/ul>\n
Organizemos os dados numa tabela para podermos relacion\u00e1-los mais facilmente.<\/p>\n \n\n\nRa\u00e7\u00e3o<\/strong><\/td>\nQuantidade di\u00e1ria
\npor animal (kg)<\/strong><\/td>\nIngredientes\u00a0(g)<\/strong><\/td>\nCusto (\u20ac)<\/strong><\/td>\n<\/tr>\n\nCarbohidratos<\/strong><\/td>\nVitaminas<\/strong><\/td>\nProte\u00ednas<\/strong><\/td>\n<\/tr>\n\nGranulado<\/strong><\/td>\n| $x$<\/td>\n | $20x$<\/td>\n | $50x$<\/td>\n | $30x$<\/td>\n | $10x$<\/td>\n<\/tr>\n | \nFarinha<\/strong><\/td>\n| $y$<\/td>\n | $50y$<\/td>\n | $10y$<\/td>\n | $30y$<\/td>\n | $5y$<\/td>\n<\/tr>\n | \nTotal<\/strong><\/td>\n| $x+y$<\/td>\n | $20x+50y$<\/td>\n | $50x+10y$<\/td>\n | $30x+30y$<\/td>\n | $10x+5y$<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n \n O problema tem as seguintes restri\u00e7\u00f5es<\/strong>:<\/p>\n\n- \n
$x\\in \\mathbb{R}_{0}^{+}$\u00a0e $y\\in \\mathbb{R}_{0}^{+}$ \nA quantidade di\u00e1ria de granulado\u00a0ou farinha por animal\u00a0\u00e9 n\u00e3o negativa;<\/div>\n<\/li>\n - \n
$20x+50y\\ge 200\\Leftrightarrow 2x+5y\\ge 20$ \nA dose m\u00ednima di\u00e1ria por animal de carbohidratos \u00e9 de 200 g;<\/div>\n<\/li>\n - $50x+10y\\ge 150\\Leftrightarrow 5x+y\\ge 15$
\nA dose m\u00ednima di\u00e1ria por animal de vitaminas \u00e9 de 150 g;<\/li>\n - $30x+30y\\ge 210\\Leftrightarrow x+y\\ge 7$
\nA dose m\u00ednima di\u00e1ria por animal de prote\u00ednas \u00e9 de 210 g.<\/li>\n<\/ol>\nCada quilograma de granulado custa 10 \u20ac e cada quilograma de farinha custa\u00a05 \u20ac.
\nPortanto, a fun\u00e7\u00e3o objetivo<\/strong> \u00e9: $C=10x+5y\\Leftrightarrow y=-2x+\\frac{C}{5}$.<\/p>\nOs pontos admiss\u00edveis, isto \u00e9, os pontos cujas coordenadas satisfazem as condi\u00e7\u00f5es impostas, s\u00e3o os pontos pertencentes ao dom\u00ednio plano representado, no seu interior ou sobre a sua fronteira.<\/p>\n Validemos as coordenadas dos pontos G e H, assinalados no referencial apresentado:<\/p>\n $\\begin{matrix}
\n\\left\\{ \\begin{array}{*{35}{l}}
\n2x+5y=20\u00a0 \\\\
\nx+y=7\u00a0 \\\\
\n\\end{array} \\right. & \\Leftrightarrow\u00a0 & \\left\\{ \\begin{array}{*{35}{l}}
\n3y=6\u00a0 \\\\
\nx+y=7\u00a0 \\\\
\n\\end{array} \\right. & \\Leftrightarrow\u00a0 & \\left\\{ \\begin{array}{*{35}{l}}
\nx=5\u00a0 \\\\
\ny=2\u00a0 \\\\
\n\\end{array} \\right. & \\to\u00a0 & H(5,2)\u00a0 \\\\
\n\\end{matrix}$<\/p>\n $\\begin{matrix}
\n\\left\\{ \\begin{array}{*{35}{l}}
\n5x+y=15\u00a0 \\\\
\nx+y=7\u00a0 \\\\
\n\\end{array} \\right. & \\Leftrightarrow\u00a0 & \\left\\{ \\begin{array}{*{35}{l}}
\n4x=8\u00a0 \\\\
\nx+y=7\u00a0 \\\\
\n\\end{array} \\right. & \\Leftrightarrow\u00a0 & \\left\\{ \\begin{array}{*{35}{l}}
\nx=2\u00a0 \\\\
\ny=5\u00a0 \\\\
\n\\end{array} \\right. & \\to\u00a0 & G(2,5)\u00a0 \\\\
\n\\end{matrix}$<\/p>\n<\/p>\n | | | | | | | | | | | |