7.
\begin{figure}[h]
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\caption{Figure 1}
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A student wants to make plastic chess pieces using a 3D printer. Figure 1 shows the central vertical cross-section of the student's design for one chess piece. The plastic chess piece is formed by rotating the region bounded by the \(y\)-axis, the \(x\)-axis, the line with equation \(x = 1\), the curve \(C _ { 1 }\) and the curve \(C _ { 2 }\) through \(360 ^ { \circ }\) about the \(y\)-axis.
The point \(A\) has coordinates ( \(1,0.5\) ) and the point \(B\) has coordinates ( \(0.5,2.5\) ) where the units are centimetres.
The curve \(C _ { 1 }\) is modelled by the equation
$$x = \frac { a } { y + b } \quad 0.5 \leqslant y \leqslant 2.5$$
- Determine the value of \(a\) and the value of \(b\) according to the model.
The curve \(C _ { 2 }\) is modelled to be an arc of the circle with centre \(( 0,3 )\).
- Use calculus to determine the volume of plastic required to make the chess piece according to the model.