We assume the box is open-top.
If the width of the box is represented by w, then its length is 2w. The area of the base is then w·2w = 2w².
The combined area of the sides is the product of the perimeter of the base, 2(w+2w) = 6w and the height, 0.5, so is 3w.
Since the base area is 1.08 m² less than the lateral area, we have ...
... 2w² = 3w -1.08
... 2w² -3w +1.08 = 0 . . . . rearrange to standard form
... w² -1.5w +0.54 = 0 . . . make the leading coefficient 1 (divide by 2)
... (w -0.9)(w -0.6) = 0 . . . factor
The width of the box may be either 0.6 meters or 0.9 meters.
The volume is the product of width, length, and height, so is ...
... V = w·2w·0.5 = w²
... V = 0.36 m³ . . . or . . . 0.81 m³. . . . . . (there are two possible answers)
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Check
For width 0.6 m, the length is 1.2 m and the perimeter is 2(0.6+1.2) = 3.6 m. The lateral area is then (0.5 m)(3.6 m) = 1.8 m². Subtracting 1.08 m² from this, we get 0.72 m². The area of the base is (0.6 m)(1.2 m) = 0.72 m², so this answer checks.
For width 0.9 m, the length is 1.8 m and the perimeter is 2(0.9+1.8) = 5.4 m. The lateral area is then (0.5m)(5.4 m) = 2.7 m². Subtracting 1.08 m² from this, we get 1.62 m². The area of the base is (0.9 m)(1.8 m) = 1.62 m², so this answer checks, too.
