3 A circular ornamental garden pond, of radius 2 metres, has weed starting to grow and cover its surface.
As the weed grows, it covers an area of \(A\) square metres. A simple model assumes that the weed grows so that the rate of increase of its area is proportional to \(A\).
3
- Show that the area covered by the weed can be modelled by
where \(B\) and \(k\) are constants and \(t\) is time in days since the weed was first noticed.
[0pt]
[4 marks]
$$A = B \mathrm { e } ^ { k t }$$
3 - When it was first noticed, the weed covered an area of \(0.25 \mathrm {~m} ^ { 2 }\). Twenty days later the weed covered an area of \(0.5 \mathrm {~m} ^ { 2 }\)
3
- State the value of \(B\).
[0pt]
[1 mark]
3
- (ii) Show that the model for the area covered by the weed can be written as
$$A = 2 ^ { \frac { t } { 20 } - 2 }$$
[4 marks]
Question 3 continues on the next page
3 - (iii) How many days does it take for the weed to cover half of the surface of the pond?
[0pt]
[2 marks]
3 - State one limitation of the model.
3 - Suggest one refinement that could be made to improve the model.
\(4 \quad \int _ { 1 } ^ { 2 } x ^ { 3 } \ln ( 2 x ) \mathrm { d } x\) can be written in the form \(p \ln 2 + q\), where \(p\) and \(q\) are rational numbers. Find \(p\) and \(q\).
[0pt]
[5 marks]