Problem $2$ CL$-$ME:$20/$Others:$25$ You want to design a cylindrical vessel $($closed at the top and bottom$)$

at a minimal cost. The next is a set of specifications the vessel should have:

$-$Total internal capacity of the vessel $=V_T$

$-$Maximum height to diameter ratio $=R_{\text{m}\text{a}\text{x}}$

$-$Minimum height to diameter ratio $=R_{\text{m}\text{i}\text{n}}$

The price of the metal used for the vessel is $P(US\frac{\text{\$}}{m^2})$

i$)$ Formulate the corresponding optimization problem to calculate the optimal dimensions of the

vessel.

ii$)$ Draw a schematic of the objective function and feasible region using the following values

$10m^3,R_{\text{m}\text{a}\text{x}}=10,R_{\text{m}\text{i}\text{n}}=1$ and $P=$$$\frac{30}{m^2}).$ Please use level curves to describe the objective

function over the feasible region.