Line of intersection of planes

The plane \(\Pi _ { 1 }\) has equation \(\mathbf { r } = 2 \mathbf { i } + \mathbf { j } + 4 \mathbf { k } + \lambda ( 2 \mathbf { i } + 3 \mathbf { j } + 4 \mathbf { k } ) + \mu ( - \mathbf { i } + \mathbf { k } )\). Obtain a cartesian equation of \(\Pi _ { 1 }\) in the form \(p x + q y + r z = d\). The plane \(\Pi _ { 2 }\) has equation \(\mathbf { r } . ( \mathbf { i } - 4 \mathbf { j } + 5 \mathbf { k } ) = 12\). Find a vector equation of the line of intersection of \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\). The line \(l\) passes through the point \(A\) with position vector \(a \mathbf { i } + ( 2 a + 1 ) \mathbf { j } - 3 \mathbf { k }\) and is parallel to \(3 c \mathbf { i } - 3 \mathbf { j } + c \mathbf { k }\), where \(a\) and \(c\) are positive constants. Given that the perpendicular distance from \(A\) to \(\Pi _ { 1 }\) is \(\frac { 15 } { \sqrt { } 6 }\) and that the acute angle between \(l\) and \(\Pi _ { 1 }\) is \(\sin ^ { - 1 } \left( \frac { 2 } { \sqrt { } 6 } \right)\), find the values of \(a\) and \(c\). \footnotetext{Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be pleased to make amends at the earliest possible opportunity.
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8 The plane \(\Pi _ { 1 }\) has equation \(\mathbf { r } = \left( \begin{array} { r } 2 \\ 3 \\ - 1 \end{array} \right) + s \left( \begin{array} { l } 1 \\ 0 \\ 1 \end{array} \right) + t \left( \begin{array} { r } 1 \\ - 1 \\ - 2 \end{array} \right)\). Find a cartesian equation of \(\Pi _ { 1 }\). The plane \(\Pi _ { 2 }\) has equation \(2 x - y + z = 10\). Find the acute angle between \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\). Find an equation of the line of intersection of \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\), giving your answer in the form \(\mathbf { r } = \mathbf { a } + \lambda \mathbf { b }\).

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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 \(b\).

1. 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.