We can use the density formula to find the volume of the paper weight, and then use the volume formula for a cube to find its dimensions.
Density formula: Density = Mass / Volume
Volume formula for a cube: Volume = Length x Width x Height
Let's start by finding the volume of the paper weight:
Density = Mass / Volume
3.487 g/cm^3 = 11,769 g / Volume
Volume = 11,769 g / 3.487 g/cm^3
Volume = 3,375.1 cm^3
Now we can use the volume formula for a cube to find its dimensions:
Volume = Length x Width x Height
3,375.1 cm^3 = L x W x H
Since the paper weight is a cube, we know that the length, width, and height are all the same, so we can call this value "x":
Volume = x^3
3,375.1 cm^3 = x^3
x = cuberoot(3,375.1)
x = 14.4 cm
Therefore, the dimensions of the paper weight are 14.4 cm x 14.4 cm x 14.4 cm (or approximately 5.7 inches x 5.7 inches x 5.7 inches).
To determine how this affects the size of the box, we need to consider the dimensions of the paper weight and add padding and space for shipping. Let's assume we add 2 cm (0.8 inches) of padding on each side of the paper weight, and we want to add an additional 2 cm (0.8 inches) of space on each side of the box for shipping. This means that the dimensions of the box would need to be:
Length = 14.4 cm + 2 cm (padding) + 2 cm (shipping) = 18.4 cm
Width = 14.4 cm + 2 cm (padding) + 2 cm (shipping) = 18.4 cm
Height = 14.4 cm + 2 cm (padding) + 2 cm (shipping) = 18.4 cm
Therefore, the box would need to be 18.4 cm x 18.4 cm x 18.4 cm (or approximately 7.2 inches x 7.2 inches x 7.2 inches) to accommodate the paper weight with padding and shipping space.