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.
(i)
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]
(ii)
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.
(iii)
Show that ${{r}^{2}}=\frac{4h}{h-4}$.
[2]
(iv)
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]
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