(i) Consider an unrestricted geometric programming problem

$$\min g(t), \quad t=\left(t_{1}, \ldots, t_{m}\right)>0,$$

where $g(t)$ is given by

$$g(t)=\sum_{i=1}^{n} c_{i} t_{1}^{a_{i 1}} \ldots t_{m}^{a_{i m}}$$

with $n \geq m$ and positive coefficients $c_{1} \ldots, c_{n}$. State the dual problem of $(*)$ and show that if $\lambda^{*}=\left(\lambda_{1}^{*}, \ldots, \lambda_{n}^{*}\right)$ is a dual optimum then any positive solution to the system

$$t_{1}^{a_{i 1}} \ldots t_{m}^{a_{i m}}=\frac{1}{c_{i}} \lambda_{i}^{*} v\left(\lambda^{*}\right), \quad i=1, \ldots, n,$$

gives an optimum for primal problem $(*)$. Here $v(\lambda)$ is the dual objective function.

(ii) An amount of ore has to be moved from a pit in an open rectangular skip which is to be ordered from a supplier.

The skip cost is $£ 36$ per $1 \mathrm{~m}^{2}$ for the bottom and two side walls and $£ 18$ per $1 \mathrm{~m}^{2}$ for the front and the back walls. The cost of loading ore into the skip is $£ 3$ per $1 \mathrm{~m}^{3}$, the cost of lifting is $£ 2$ per $1 \mathrm{~m}^{3}$, and the cost of unloading is $£ 1$ per $1 \mathrm{~m}^{3}$. The cost of moving an empty skip is negligible.

Write down an unconstrained geometric programming problem for the optimal size (length, width, height) of skip minimizing the cost of moving $48 \mathrm{~m}^{3}$ of ore. By considering the dual problem, or otherwise, find the optimal cost and the optimal size of the skip.