N2014 P1 Q11

[It is given that the volume of a sphere of radius \(r\) is \(\frac{4}{3}\pi {{r}^{3}}\) and that the volume of a circular cone with base radius \(r\) and height \(h\) is \(\frac{1}{3}\pi {{r}^{2}}h\).]

A toy manufacturer makes a toy which consists of a hemisphere of radius \(r\) cm joined to a circular cone of base radius \(r\) cm and height \(h\) cm (see diagram). The manufacturer determines that the length of the slant edge of the cone must be \(4\) cm and that the total volume of the toy, \(V\) cm\(^{3}\), should be as large as possible.

1. Find a formula for \(V\) in terms of \(r\). Given that \(r={{r}_{1}}\) is the value of \(r\) which gives the maximum value of \(V\), show that \({{r}_{1}}\) satisfies the equation \(45{{r}^{4}}-768{{r}^{2}}+1024=0\).[6]
2. Find the two solutions to the equation in part (i) for which \(r>0\), giving your answers correct to \(3\) decimal places.[2]
3. Show that one of the solutions found in part (ii) does not give a stationary value of \(V\). Hence write down the value of \({{r}_{1}}\) and find the corresponding value of \(h\).[3]
4. Sketch the graph showing the volume of the toy as the radius of the hemisphere varies.[3]