Methods2023 Exam 2 Section B · Q2CAS active11 marks

Observation wheel height model

The following diagram represents an observation wheel, with its centre at point $P$ . Passengers are seated in pods, which are carried around as the wheel turns. The wheel moves anticlockwise with constant speed and completes one full rotation every 30 minutes. When a pod is at the lowest point of the wheel (point $A$ ), it is 15 metres above the ground. The wheel has a radius of 60 metres.

Consider the function $h(t) = -60\cos(bt) + c$ for some , which models the height above the ground of a pod originally situated at point , after time minutes.

Show that $b = \frac{\pi}{15}$ and $c = 75$ .

2 marks

Hints, answer checking and worked solutions come with a Season Pass.

Find the average height of a pod on the wheel as it travels from point $A$ to point $B$ .

Give your answer in metres, correct to two decimal places.

2 marks

Hints, answer checking and worked solutions come with a Season Pass.

Find the average rate of change, in metres per minute, of the height of a pod on the wheel as it travels from point $A$ to point $B$ .

1 mark

Hints, answer checking and worked solutions come with a Season Pass.

d.i.

After 15 minutes, the wheel stops moving and remains stationary for 5 minutes. After this, it continues moving at double its previous speed for another 7.5 minutes.

The height above the ground of a pod that was initially at point $A$ , after $t$ minutes, can be modelled by the piecewise function $w$ :

$w(t) = \begin{cases} h(t) & 0 \le t < 15 \\ k & 15 \le t < 20 \\ h(mt + n) & 20 \le t \le 27.5 \end{cases}$

where $k \ge 0$ , $m \ge 0$ and $n \in \R$ .

State the values of $k$ and $m$ .

1 mark

Hints, answer checking and worked solutions come with a Season Pass.

Methods2023 Exam 2 Section B · Q2CAS active11 marks

Observation wheel height model

Consider the function $h(t) = -60\cos(bt) + c$ for some , which models the height above the ground of a pod originally situated at point , after time minutes.

Show that $b = \frac{\pi}{15}$ and $c = 75$ .

2 marks

Hints, answer checking and worked solutions come with a Season Pass.

Find the average height of a pod on the wheel as it travels from point $A$ to point $B$ .

Give your answer in metres, correct to two decimal places.

2 marks

Hints, answer checking and worked solutions come with a Season Pass.

Find the average rate of change, in metres per minute, of the height of a pod on the wheel as it travels from point $A$ to point $B$ .

1 mark

Hints, answer checking and worked solutions come with a Season Pass.

d.i.

After 15 minutes, the wheel stops moving and remains stationary for 5 minutes. After this, it continues moving at double its previous speed for another 7.5 minutes.

The height above the ground of a pod that was initially at point $A$ , after $t$ minutes, can be modelled by the piecewise function $w$ :

$w(t) = \begin{cases} h(t) & 0 \le t < 15 \\ k & 15 \le t < 20 \\ h(mt + n) & 20 \le t \le 27.5 \end{cases}$

where $k \ge 0$ , $m \ge 0$ and $n \in \R$ .

State the values of $k$ and $m$ .

1 mark

Hints, answer checking and worked solutions come with a Season Pass.