Considere a reta que passa por O, sendo $0 < \\alpha\u00a0 < \\frac{\\pi }{2}$, e que encontra as paredes em A e B.<\/p>\n d) Pretende-se transportar, naquele corredor, um painel, em posi\u00e7\u00e3o vertical. b) Para $\\alpha\u00a0 \\in \\left] {0,\\frac{\\pi }{2}} \\right[$, temos: \\[\\begin{array}{*{20}{l}}![\"\"](\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/12pag128-11.jpg\" "\\"Corredor\\"")
<\/a>Na figura est\u00e1 representado um corredor de um museu.<\/p>\n\n
\nQual as consequ\u00eancias pr\u00e1ticas que se podem tirar do estudo feito nas al\u00edneas anteriores?<\/li>\n<\/ol>\nResolu\u00e7\u00e3o >><\/a><\/span><\/div><\/div>\n\n\nResolu\u00e7\u00e3o<\/b><\/span><\/p>\n\n
Ora,
\n$$\\operatorname{sen} \\alpha\u00a0 = \\frac{5}{{\\overline {OA} }} \\Leftrightarrow \\overline {OA}\u00a0 = \\frac{5}{{\\operatorname{sen} \\alpha }}$$
\n\u00ad<\/li>\n
\n$$\\operatorname{sen} \\left( {\\frac{\\pi }{2} – \\alpha } \\right) = \\frac{1}{{\\overline {OB} }} \\Leftrightarrow \\overline {OB}\u00a0 = \\frac{1}{{\\operatorname{sen} \\left( {\\frac{\\pi }{2} – \\alpha } \\right)}} \\Leftrightarrow \\overline {OB}\u00a0 = \\frac{1}{{\\cos \\alpha }}$$
\n\u00ad<\/li>\n
\n{f(\\alpha )}& = &{\\overline {OA}\u00a0 + \\overline {OB} } \\\\
\n{}& = &{\\frac{5}{{\\operatorname{sen} \\alpha }} + \\frac{1}{{\\cos \\alpha }}}
\n\\end{array}$$<\/p>\n
\n{f’\\left( \\alpha\u00a0 \\right)}& = &{{{\\left( {\\frac{5}{{\\operatorname{sen} \\alpha }} + \\frac{1}{{\\cos \\alpha }}} \\right)}^\\prime }} \\\\
\n{}& = &{\\frac{{ – 5\\cos \\alpha }}{{{{\\operatorname{sen} }^2}\\alpha }} + \\frac{{\\operatorname{sen} \\alpha }}{{{{\\cos }^2}\\alpha }}} \\\\
\n{}& = &{\\frac{{\\operatorname{sen} \\alpha }}{{{{\\cos }^2}\\alpha }} – \\frac{{5\\cos \\alpha }}{{{{\\operatorname{sen} }^2}\\alpha }}}
\n\\end{array}\\]<\/p>\n