Answers:
(a) The ledge can support the plant box. It is safe to put it there.
(b) No the density would not change.
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Explanation for part (a)
We'll need the volume of the cube.
volume = side*side*side
volume = 13*13*13
volume = 2,197
Or you could do
volume = (side)^3
volume = (13)^3
volume = 2,197
The volume of the cube is 2,197 cubic cm. Multiply this with the density to determine its weight.
1.33*2197 = 2,922.01
The soil weighs 2,922.01 grams which is below the 2400 gram weight limit. Therefore, the ledge can support the planter.
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Explanation for part (b)
The weight would go down if you only put in half the soil, but the density would not change regardless of how much soil is in there. Why doesn't the density change? Because the density is the amount of stuff per unit volume. The density tells us how packed in a material is. We can scale the amount of material up or down, and retain the same density.
This is based off physics concepts which are bit beyond the scope of this math course, but different materials have different densities. It's common to use a reference table to look up densities of various materials. For instance, ice is less dense than water. This is why ice floats rather than sinks. As another example, gold is more dense than iron. So 1 cubic unit of gold will weigh more compared to 1 cubic unit of iron.
