2022 RI Promo Q11

[It is given that a right circular cone of radius \(r\) and height \(h\) has total surface area \(\pi {{r}^{2}}+\pi r\sqrt{{{r}^{2}}+{{h}^{2}}}\) and volume \(\frac{1}{3}\pi {{r}^{2}}h\).]

An ice-cream company plans to launch a new product called Tasty Cone into the market. Each Tasty Cone is in the shape of a right circular cone with a base radius \(r\) cm and height \(h\) cm as shown in Fig 1.

1. The volume of each Tasty Cone is set by the company to be \(100\) ml and the total surface area, \(S\)cm\(^{2}\), should be as small as possible to reduce the cost of packaging. It is given that \(1\)ml\(=1\)cm\(^{3}\).
   Show that \(S=\pi {{r}^{2}}+\frac{1}{r}\sqrt{{{\pi }^{2}}{{r}^{6}}+90000}\).[2]
2. Sketch the graph of \(S\) for \(r>0\). Hence write down the value of \(r\) and find the corresponding value of \(h\) which will give the smallest \(S\), giving your answers correct to \(3\) significant figures.[3]

A special edition of the Tasty Cone will include a mystery flavour in the shape of a sphere inscribed in the cone as shown in Fig. 2. The sphere has centre \(O\) and radius \(2\) cm. The cross-section of the cone is shown in Fig. 3, where \(A\) is a point of contact of the sphere with the cone.

3. Show that \({{r}^{2}}=\frac{4h}{h-4}\).[2]
4. Use differentiation to find the exact values of \(h\) and \(r\) which will give the smallest volume of the cone in the special edition.[5]