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Compreender as propriedades das funções trigonométricas. Calcular expressões e resolver equações envolvendo funções trigonométricas e funções trigonométricas inversas. Compreender a taxa de variação quando as grandezas dependem umas das outras. Aplicar a regra da cadeia para relacionar grandezas expressas em unidades diferentes.
Um barco navega
diretamente em direção a um arranha-céu de \(100\unit {m}\) que fica à
beira de um porto. Um observador está em pé na proa do barco. Seja \(\theta \) o ângulo
formado pelas retas que ligam o topo e a base do arranha-céu à cabeça do observador
(para todos os efeitos, você pode supor que a reta que parte da base do arranha-céu e
o contorno do arranha-céu são perpendiculares entre si). Seja \(x\) a distância horizontal
do barco ao prédio. Expresse o ângulo \(\theta \) como função de \(x\). \[ \theta (x)=\class{xm-answer}{\cssId{xm-answer-1}{\phantom{\rule{7em}{1.5em}}}}\unit {rad} \]
Agora suponha que o
barco esteja navegando diretamente em direção ao arranha-céu a \(3\unit {m/s}\). Isso significa que \[ \frac {dx}{dt}=\class{xm-answer}{\cssId{xm-answer-2}{\phantom{\rule{7em}{1.5em}}}}\unit {m/s} \]
Usando suas respostas anteriores, determine \(\frac {d\theta }{dt}\) quando o barco está a \(x=300\unit {m}\) do prédio. \[ \frac {d\theta }{dt}=\class{xm-answer}{\cssId{xm-answer-3}{\phantom{\rule{7em}{1.5em}}}}\unit {rad/s} \]