Applications of Circular Functions - Questions
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Question 1
20992
The formula \(h\left( t \right) = 3\cos \left( {\dfrac{\pi }{6}t} \right)\) for \(0 \le t \le 24\), represents the height, \(h(t)\) metres above the mean sea level, where \(t\) is the number of hours after midnight. A boat can only cross the harbour when the tide is at least \(1.5\) metres above the sea-level. At what times can the boat cross the harbour?
\(\begin{array}{*{20}{l}}{midnight{\rm{ }} < t < {\rm{ }}2 \,am}\\{10\, am{\rm{ }} < t < {\rm{ }}2\, pm}\\{10\, pm{\rm{ }} < t < {\rm{ }}midnight\;}\end{array}\)
| \begin{align} &h(t)=3 \cos \left(\frac{\pi}{6} t\right) \\ &\begin{aligned} \text { Let }\;\; 3 \cos \left(\frac{\pi}{6} t\right) &=1.5 \\ \left.\cos( \frac{\pi}{6} t\right) &=\frac{1}{2} \\ \frac{\pi}{6} t &=\cos ^{-1}\left(\frac{1}{2}\right) \\ \frac{\pi}{6} t &=\frac{\pi}{3}, \frac{5 \pi}{3}, \frac{7 \pi}{3}, \frac{11 \pi}{3}\\ \therefore t&=2,10,14,22 \end{aligned}\\ &\text{From the graph:}\\ &\begin{aligned} \text{midnight } &<t<2 \mathrm{~am}\\ 10 \mathrm{~am}&<t<2 \mathrm{~pm}\\ 10 \mathrm{~pm}&<t< \text{ midnight} \end{aligned} \end{align} |
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