6
2
\end{array} \right)$$
where \(\mu\) is a scalar parameter.
The plane \(\Pi _ { 2 }\) contains the line \(l _ { 1 }\) and the line \(l _ { 2 }\)
(c) Determine a vector equation for the line of intersection of \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\)
The plane \(\Pi _ { 3 }\) has equation r. \(\left( \begin{array} { l } 1
1
a \end{array} \right) = b\) where \(a\) and \(b\) are constants.
Given that the planes \(\Pi _ { 1 } , \Pi _ { 2 }\) and \(\Pi _ { 3 }\) form a sheaf,
(d) determine the value of \(a\) and the value of \(b\).
- Water is flowing into and out of a large tank.
Initially the tank contains 10 litres of water.
The rate of flow of the water is modelled so that
- there are \(V\) litres of water in the tank at time \(t\) minutes after the water begins to flow
- water enters the tank at a rate of \(\left( 3 - \frac { 4 } { 1 + \mathrm { e } ^ { 0.8 t } } \right)\) litres per minute
- water leaves the tank at a rate proportional to the volume of water remaining in the tank
Given that when \(t = 0\) the volume of water in the tank is decreasing at a rate of 3 litres per minute, use the model to
(a) show that the volume of water in the tank at time \(t\) satisfies
$$\frac { \mathrm { d } V } { \mathrm {~d} t } = 3 - \frac { 4 } { 1 + \mathrm { e } ^ { 0.8 t } } - 0.4 V$$
(b) Determine \(\frac { \mathrm { d } } { \mathrm { d } t } \left( \arctan \mathrm { e } ^ { 0.4 t } \right)\)
Hence, by solving the differential equation from part (a),
(c) determine an equation for the volume of water in the tank at time \(t\).
Give your answer in simplest form as \(V = \mathrm { f } ( t )\)
After 10 minutes, the volume of water in the tank was 8 litres.
(d) Evaluate the model in light of this information.