Suppose you are a materials engineer working for a beverage company and have been asked to identify what material should be used to minimize the cost of fabricating soda containers. For ergonomic reasons the company has decided to use a cylindrical container with a radius of R = 3.25cm, and a height of H = 12cm. Assume that the top and bottom of the container are to have the same thickness as the sidewalls. Also, for simplicity, assume that the container opening can be added at no additional cost after the container has been made (thus, the container is assumed to be a hollow cylinder closed at its ends). A diagram of the cylindrical container is provided below in Figure 1a.

The maximum stress on the walls of the cylindrical container will be the hoop stress, σh. The hoop stress acts in the circumferential direction and is shown in Figure 1b. The equation for the hoop stress is given as σh = PR/t, where t is the thickness of the walls of the container and P and R are the internal pressure and radius of the container, respectively. Given that the maximum pressure exerted by carbonated beverages on the internal walls of the container is 500kPa, and that the container must have a safety factor of two such that the condition 𝜎h < 2·𝜎y (where 𝜎y is the yield strength of the material) must hold: 

i. Derive an expression for the cost of one container in terms of the price of the material it is fabricated from, pm, which is available in units of [$/Kg]. ( Note: the volume of a hollow cylinder ~ 2Ï€RHt when t is small compared to R ).