11 The height, in metres, of the sea at a coastal town during a day may be modelled by the function
$$\mathrm { f } ( t ) = 1.7 + 0.8 \sin ( 30 t ) ^ { \circ }$$
where \(t\) is the number of hours after midnight.
- (a) Find the maximum height of the sea as given by this model.
(b) Find the time of day at which this maximum height first occurs. - Determine the time when, according to this model, the height of the sea will first be 1.2 m .
The height, in metres, at a different coastal town during a day may be modelled by the function
$$\mathrm { g } ( t ) = a + b \sin ( c t + d ) ^ { \circ }$$
where \(t\) is the number of hours after midnight.
- It is given that at this different coastal town the maximum height of the sea is 3.1 m , and this height occurs at 0500 and 1700. The minimum height of the sea is 0.7 m , and this height occurs at 1100 and 2300 . Find the values of the constants \(a , b , c\) and \(d\).
- It is instead given that the maximum height of the sea actually occurs at 0500 and 1709 . State, with a reason, how this will affect the value of \(c\) found in part (iii).
\includegraphics[max width=\textwidth, alt={}, center]{74a37bca-0b28-4c48-bd21-a9304f31b8f8-6_563_568_322_751}
The diagram shows the curve \(y = \mathrm { e } ^ { \sqrt { x + 1 } }\) for \(x \geqslant 0\). - Use the substitution \(u ^ { 2 } = x + 1\) to find \(\int \mathrm { e } ^ { \sqrt { x + 1 } } \mathrm {~d} x\).
- Make \(x\) the subject of the equation \(y = \mathrm { e } ^ { \sqrt { x + 1 } }\).
- Hence show that \(\int _ { \mathrm { e } } ^ { \mathrm { e } ^ { 4 } } \left( ( \ln y ) ^ { 2 } - 1 \right) \mathrm { d } y = 9 \mathrm { e } ^ { 4 }\).
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