4.
\begin{figure}[h]
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\caption{Figure 2}
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\end{figure}
Figure 2 shows a sketch of the curve \(C\) with equation \(y = \frac { 4 } { x - 3 } , x \neq 3\).
The points \(A\) and \(B\) on the curve have \(x\)-coordinates 3.25 and 5 respectively.
- Write down the \(y\)-coordinates of \(A\) and \(B\).
- Show that an equation of \(C\) is \(\frac { 3 y + 4 } { y } , y \neq 0\).
The shaded region \(R\) is bounded by \(C\), the \(y\)-axis and the lines through \(A\) and \(B\) parallel to the \(x\) axis. The region \(R\) is rotated through \(360 ^ { \circ }\) about the \(y\)-axis to form a solid shape \(S\).
- Find the volume of \(S\), giving your answer in the form \(\pi ( a + b \ln c )\), where \(a , b\) and \(c\) are integers.
The solid shape \(S\) is used to model a cooling tower. Given that 1 unit on each axis represents 3 metres,
- show that the volume of the tower is approximately \(15500 \mathrm {~m} ^ { 3 }\).