Parametric equations in modellingAQA A-Level Maths: Mind map

Exam questions on Parametric equations in modelling

1. A ball is kicked from level ground. At time $t$ seconds after the kick, its horizontal distance from the starting point is $x=12t$ metres and its height is $y=16t-5t^2$ metres. The model applies until the ball lands.
   Find the greatest height reached by the ball.2 marks
2. A Ferris wheel has its centre $12$ m above the ground. The position of a passenger pod is modelled by $x=10\sin(0.2t)$, $y=12-10\cos(0.2t)$, where $x$ is the horizontal distance in metres from the vertical line through the centre, $y$ is the height above the ground in metres and $t$ is the time in seconds after the pod passes its lowest point. The angle $0.2t$ is in radians.
   State the greatest height of the pod above the ground, and justify your answer.2 marks
3. A runner follows an oval track. Relative to the centre of the track, the runner's position at time $t$ seconds is $x=80\cos\theta$, $y=50\sin\theta$, where $\theta=0.1t$ radians and $x$ and $y$ are in metres.
   Show that the runner's path has the Cartesian equation $\dfrac{x^2}{6400}+\dfrac{y^2}{2500}=1$.3 marks