Example 3 Construct a rectangular open pool with a volume of $8 \mathrm{~m}^{3}$ and a depth of $2 \mathrm{~m}$. If the cost of the pool bottom and the pool walls are 120 yuan $/ \mathrm{m}^{2}$ and 80 yuan $/ \mathrm{m}^{2}$ respectively, write the function expression of the total cost $y$ (yuan) in terms of the length of one side of the bottom $x(\mathrm{~m})$, and specify its domain and minimum value.
From the given information, the base area is $4 \mathrm{~m}^{2}$, and the lengths of the two sides of the base are $x \mathrm{~m}$ and $\frac{4}{x} \mathrm{~m}$. The total cost is
$$
y=2 \times\left(x+\frac{4}{x}\right) \times 2 \times 80+120 \times 4,
$$

which simplifies to $y=320\left(x+\frac{4}{x}\right)+480$, with the domain being $x>0$. Furthermore, $y \geqslant 320 \times 2 \times 2+480=1760$, so when $x=2(\mathrm{~m})$, $y$ reaches its minimum value of 1760 (yuan).