A thermal storage chest with a square base and a box-like shape must hold $20{\text{ ft}}^3$ of material. The bottom panel is made of reinforced insulation, which is three times more expensive per square foot than the side panels, while the top lid is constructed from a lightweight material that costs the same as the sides. Determine the side length $s$ of the square base and the height $h$ of the storage chest that will minimize the total material cost.