Moderate -0.8 This question requires recognizing that 'rate of change of height' means dh/dt, writing dh/dt = a cos(kt), then using given information to find constants: a = 1.3 (maximum of cosine is 1) and k = 2π/12 = π/6 (period is 12 hours). It's straightforward application of definitions with minimal calculation, easier than average A-level questions.
7 The height of the tide in a certain harbour is \(h\) metres at time \(t\) hours. Successive high tides occur every 12 hours.
The rate of change of the height of the tide can be modelled by a function of the form \(a \cos ( k t )\), where \(a\) and \(k\) are constants. The largest value of this rate of change is 1.3 metres per hour.
Write down a differential equation in the variables \(h\) and \(t\). State the values of the constants \(a\) and \(k\).
7 The height of the tide in a certain harbour is $h$ metres at time $t$ hours. Successive high tides occur every 12 hours.
The rate of change of the height of the tide can be modelled by a function of the form $a \cos ( k t )$, where $a$ and $k$ are constants. The largest value of this rate of change is 1.3 metres per hour.
Write down a differential equation in the variables $h$ and $t$. State the values of the constants $a$ and $k$.
\hfill \mbox{\textit{AQA C4 2013 Q7 [3]}}