- The curve \(C\) has equation
$$r = a ( p + 2 \cos \theta ) \quad 0 \leqslant \theta < 2 \pi$$
where \(a\) and \(p\) are positive constants and \(p > 2\)
There are exactly four points on \(C\) where the tangent is perpendicular to the initial line.
- Show that the range of possible values for \(p\) is
$$2 < p < 4$$
- Sketch the curve with equation
$$r = a ( 3 + 2 \cos \theta ) \quad 0 \leqslant \theta < 2 \pi \quad \text { where } a > 0$$
John digs a hole in his garden in order to make a pond.
The pond has a uniform horizontal cross section that is modelled by the curve with equation
$$r = 20 ( 3 + 2 \cos \theta ) \quad 0 \leqslant \theta < 2 \pi$$
where \(r\) is measured in centimetres.
The depth of the pond is 90 centimetres.
Water flows through a hosepipe into the pond at a rate of 50 litres per minute.
Given that the pond is initially empty, - determine how long it will take to completely fill the pond with water using the hosepipe, according to the model. Give your answer to the nearest minute.
- State a limitation of the model.