The length of the box is 6x inches
The width of the box is 1/3 of the length, that is:
1/3 * 6x = 2x inches
The height of the box is 1/2 of the length, that is:
1/2 * 6x = 3x inches
The volume of the box is 972 cubic inches.
The volume of a box (rectangular box) is given as:
V = L * W * H
where L = length, W = width, H = height
This means that:
V = 6x * 2x * 3x
[tex]\begin{gathered} 972=36x^3 \\ \Rightarrow x^3\text{ = }\frac{972}{36} \\ x^3\text{ = 27} \\ \Rightarrow\text{ x = }\sqrt[3]{27} \\ x\text{ = 3} \end{gathered}[/tex]