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Uma janela deve ser construída com o formato de um semicírculo sobre um
retângulo:
Se o perímetro externo mede \(10\) pés,
maximize a área. Arredonde sua resposta para o décimo de pé quadrado mais
próximo.
xmhintVamos colocar rótulos na figura.
xmhintO perímetro externo, \(P\), é dado por
\(P=2r+2x+r\pi \).
Como sabemos que \(P=10\), podemos expressar \(x\) em termos de \(r\).
\(x=5-r\frac {\pi }{2}-r\).
A área, A , é a soma da área do retângulo com a área do semicírculo:
\(A= 2\cdot r\cdot x+\frac {\pi }{2} r^2\)
xmhintAgora podemos expressar a área \(A\) como função de \(r\):
\(A(r)=2\cdot r\cdot (5-r\frac {\pi }{2}-r)+\frac {\pi }{2} r^2=10r-\frac {\pi }{2}r^2-2r^2\).
O domínio de A é o intervalo fechado \(\left [0,\frac {10}{2+\pi }\right ]\). (Lembre-se: P=10!)
xmhintAgora precisamos
encontrar o máximo absoluto de \(A\) em seu domínio. Primeiro, precisamos encontrar os
pontos críticos de \(A\). Como
\(A'(r)=10-\pi r-4r\),
segue que a função \(A\) tem seu único ponto crítico em \(r=\frac {\class{xm-answer}{\cssId{xm-answer-1}{\phantom{\rule{7em}{1.5em}}}}}{4+\pi }\) .
xmhintAgora, calculamos \(A\) nas
extremidades e no ponto crítico e, em seguida, comparamos os valores.
\(A(0)=0\)
\(A\left (\frac {10}{2+\pi }\right )=\frac {50\pi }{(2+\pi )^2}\)
\(A\left (\frac {\class{xm-answer}{\cssId{xm-answer-2}{\phantom{\rule{7em}{1.5em}}}}}{4+\pi }\right )=\frac {\class{xm-answer}{\cssId{xm-answer-3}{\phantom{\rule{7em}{1.5em}}}}}{4+\pi }\approx 7\)
Isso parece complicado. É melhor argumentar que a derivada satisfaz
\(A'(r)>0\) em \(\left (0,\frac {10}{4+\pi }\right )\) e \(A'(r)<0\) em \(\left (\frac {10}{4+\pi },\frac {10}{2+\pi }\right )\),
pois \(A'(r)=10-\pi r-4r=10-r(\pi +4)=-(\pi +4)\left (r-\frac {10}{\pi +4}\right )\).
Isso implica que o máximo absoluto ocorre no ponto crítico.
\[ A = \class{xm-answer}{\cssId{xm-answer-4}{\phantom{\rule{7em}{1.5em}}}} ft^2 \]
![[Picture]](/rendered/optimization/exercises/singVarCalcOptimization13-a8e0f434b2679c699ecbb68123942b09.svg)
![[Picture]](/rendered/optimization/exercises/singVarCalcOptimization13-619e13cbc7ff9a3469a088f0659e8afc.svg)