7. In a chemical reaction two substances combine to form a third substance. At time \(t , t \geq 0\), the concentration of this third substance is \(x\) and the reaction is modelled by the differential equation
$$\frac { \mathrm { d } x } { \mathrm {~d} t } = k ( 1 - 2 x ) ( 1 - 4 x ) , \text { where } k \text { is a positive constant. }$$
- Solve this differential equation and hence show that
$$\ln \left| \frac { 1 - 2 x } { 1 - 4 x } \right| = 2 k t + c , \text { where } c \text { is an arbitrary constant. }$$
- Given that \(x = 0\) when \(t = 0\), find an expression for \(x\) in terms of \(k\) and \(t\).
- Find the limiting value of the concentration \(x\) as \(t\) becomes very large.
\section*{8.}
\begin{figure}[h]
\captionsetup{labelformat=empty}
\caption{Figure 2}
\includegraphics[alt={},max width=\textwidth]{615ec68b-3a32-4309-bb54-acf39ed09f96-05_716_1026_326_468}
\end{figure}
Part of the design of a stained glass window is shown in Fig. 2. The two loops enclose an area of blue glass. The remaining area within the rectangle \(A B C D\) is red glass.
The loops are described by the curve with parametric equations
$$x = 3 \cos t , \quad y = 9 \sin 2 t , \quad 0 \leq t < 2 \pi .$$ - Find the cartesian equation of the curve in the form \(y ^ { 2 } = \mathrm { f } ( x )\).
- Show that the shaded area in Fig. 2, enclosed by the curve and the \(x\)-axis, is given by
$$\int _ { 0 } ^ { \frac { \pi } { 2 } } A \sin 2 t \sin t \mathrm {~d} t , \text { stating the value of the constant } A \text {. }$$
- Find the value of this integral.
The sides of the rectangle \(A B C D\), in Fig. 2, are the tangents to the curve that are parallel to the coordinate axes. Given that 1 unit on each axis represents 1 cm ,
- find the total area of the red glass.