7
0
7
\end{array} \right) + \mu \left( \begin{array} { r }
3
- 5
4
\end{array} \right) , \quad \text { where } \mu \text { is a scalar parameter. }$$
- Find the value of the constant \(p\).
- Show that \(l _ { 1 }\) and \(l _ { 2 }\) intersect and find the position vector of their point of intersection, \(C\).
- Find the size of the angle \(A C B\), giving your answer in degrees to 3 significant figures.
- Find the area of the triangle \(A B C\), giving your answer to 3 significant figures.
7. The rate of increase of the number, \(N\), of fish in a lake is modelled by the differential equation
$$\frac { \mathrm { d } N } { \mathrm {~d} t } = \frac { ( k t - 1 ) ( 5000 - N ) } { t } \quad t > 0 , \quad 0 < N < 5000$$
In the given equation, the time \(t\) is measured in years from the start of January 2000 and \(k\) is a positive constant. - By solving the differential equation, show that
$$N = 5000 - A t \mathrm { e } ^ { - k t }$$
where \(A\) is a positive constant.
After one year, at the start of January 2001, there are 1200 fish in the lake.
After two years, at the start of January 2002, there are 1800 fish in the lake.
- Find the exact value of the constant \(A\) and the exact value of the constant \(k\).
- Hence find the number of fish in the lake after five years. Give your answer to the nearest hundred fish.