2022 RI P1 Q11

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

Globally, \(2\) trillion drink cans are produced every year. Drink cans constitute part of the packaging cost for beverage companies and using the appropriate material in appropriate amounts will enable them to save costs. A new beverage company wants to produce drink cans using a particular material for the top and bottom of a cylindrical shaped drink can, and another material for the curved body. For a fixed thickness, the material for the top and bottom costs \(\$1.20/\)m\(^{2}\) and the material for the curved side costs \(\$0.90\)/m\(^{2}\).

1. For a fixed volume of \(100\pi \) ml per can, show, using differentiation, that the radius \(r\) of the most economical can is approximately \(3.35\)cm and evaluate the corresponding height \(h\).[7]
2. Hence, find the cost of the most economical can, giving your answer correct to the nearest \(0.1\) cent.[1]

In order to make the can more attractive, the company redesigns the can such that the base radius of the can is \(3\) cm and the height of the cylindrical part is \(10\) cm. At the top of each can, there is a tapering in before being covered by a circular lid of radius \(1.5\)cm, with the dimensions of the can as shown below.

3. The beverage is pumped into the can at a rate of \(90\pi \)ml/s. Find the rate at which the liquid level in the can is increasing when it is \(1\) cm from the lid of the can.[6]