Standard +0.8 This question combines integration by parts with a separable differential equation requiring careful algebraic manipulation. Part (a) is routine, but part (b) requires recognizing the separation method, integrating both sides (using part (a)), applying initial conditions, and solving for x at t=45—a multi-step problem requiring more synthesis than typical C4 questions.
\(\quad\) Find \(\int t \cos \left( \frac { \pi } { 4 } t \right) \mathrm { d } t\).
The platform of a theme park ride oscillates vertically. For the first 75 seconds of the ride,
$$\frac { \mathrm { d } x } { \mathrm {~d} t } = \frac { t \cos \left( \frac { \pi } { 4 } t \right) } { 32 x }$$
where \(x\) metres is the height of the platform above the ground after time \(t\) seconds.
At \(t = 0\), the height of the platform above the ground is 4 metres.
Find the height of the platform after 45 seconds, giving your answer to the nearest centimetre.
(6 marks)
8
\begin{enumerate}[label=(\alph*)]
\item $\quad$ Find $\int t \cos \left( \frac { \pi } { 4 } t \right) \mathrm { d } t$.
\item The platform of a theme park ride oscillates vertically. For the first 75 seconds of the ride,
$$\frac { \mathrm { d } x } { \mathrm {~d} t } = \frac { t \cos \left( \frac { \pi } { 4 } t \right) } { 32 x }$$
where $x$ metres is the height of the platform above the ground after time $t$ seconds.\\
At $t = 0$, the height of the platform above the ground is 4 metres.\\
Find the height of the platform after 45 seconds, giving your answer to the nearest centimetre.\\
(6 marks)
\end{enumerate}
\hfill \mbox{\textit{AQA C4 2013 Q8 [10]}}