Standard +0.3 This is a straightforward first-order separable differential equation with standard integration techniques. Part (a) is routine integration, part (b) involves separating variables and applying initial conditions (standard procedure), part (c) requires finding the maximum of an exponential function (straightforward calculus), and part (d) involves solving a simple trigonometric equation. All steps are textbook procedures with no novel insight required, making it slightly easier than average.
10. a. Find \(\int \frac { 1 } { 30 } \cos \frac { \pi } { 6 } t \mathrm {~d} t\).
The height above ground, \(X\) metres, of the passenger on a wooden roller coaster can be modelled by the differential equation
$$\frac { d \mathrm { X } } { \mathrm {~d} t } = \frac { 1 } { 30 } X \cos \left( \frac { \pi } { 6 } t \right)$$
where \(t\) is the time, in seconds, from the start of the ride.
At time \(t = 0\), the passenger is 6 m above the ground.
b. Show that \(X = k e ^ { \frac { 1 } { 5 \pi } \sin \left( \frac { \pi } { 6 } t \right) }\) where the value of the constant \(k\) should be found.
c. Show that the maximum height of the passenger above the ground is 6.39 m .
The passenger reaches the maximum height, for the second time, \(T\) seconds after the start of the ride.
d. Find the value of \(T\).
For identifying that it would reach the maximum height for the second time when \(\frac{\pi}{6}t = \frac{5\pi}{2}\)
### Part a:
$\frac{1}{5\pi}\sin\frac{\pi}{6}t + c$ | A1 | Correct integration
| M1 | $\int \frac{1}{30}\cos\frac{\pi}{6}t \, dt = A\sin\frac{\pi}{6}t + c$
### Part b:
$X = 6e^{\frac{1}{5\pi}\sin(\frac{\pi}{6}t)}$ | A1 | Finds $k$
| M1 | Substitutes $t = 0, X = 6$
| M1 | Rearranges to have $X = \cdots$
| M1 | Integrates both sides
| A1 | Integrates both sides
| M1 | separation of variables
### Part c:
Maximum height is 6.39 m | B1 | shows that the maximum height is 6.39 m.
### Part d:
$t = 15$ | A1 |
| M1 | For identifying that it would reach the maximum height for the second time when $\frac{\pi}{6}t = \frac{5\pi}{2}$
10. a. Find $\int \frac { 1 } { 30 } \cos \frac { \pi } { 6 } t \mathrm {~d} t$.
The height above ground, $X$ metres, of the passenger on a wooden roller coaster can be modelled by the differential equation
$$\frac { d \mathrm { X } } { \mathrm {~d} t } = \frac { 1 } { 30 } X \cos \left( \frac { \pi } { 6 } t \right)$$
where $t$ is the time, in seconds, from the start of the ride.\\
At time $t = 0$, the passenger is 6 m above the ground.\\
b. Show that $X = k e ^ { \frac { 1 } { 5 \pi } \sin \left( \frac { \pi } { 6 } t \right) }$ where the value of the constant $k$ should be found.\\
c. Show that the maximum height of the passenger above the ground is 6.39 m .
The passenger reaches the maximum height, for the second time, $T$ seconds after the start of the ride.\\
d. Find the value of $T$.\\
\hfill \mbox{\textit{Edexcel PMT Mocks Q10 [9]}}