A cube of compressible material (such as Styrofoam or balsa...

A cube of compressible material (such as Styrofoam or balsa wood) has a density $\rho$ and sides of length $L .$ (a) If you keep its mass the same, but compress each side to half its length, what will be its new density, in terms of $\rho ?$ (b) If you keep the mass and shape the same, what would the length of each side have to be (in terms of $L )$ so that the density of the cube was three times its original value?

Key Concepts

Scaling Laws

Scaling laws describe how geometric properties transform under changes in size. When the dimensions of an object are scaled, properties like volume and density adjust according to specific mathematical relationships, which is vital in solving problems involving compressed or expanded objects.

Mass Conservation

In situations where the total mass of an object remains constant, any alteration in the object's volume will directly change its density. A decrease in volume leads to an increase in density, and vice versa, assuming mass does not change.

Volume of a Cube

The volume of a cube is calculated by raising its side length to the third power. This cubic relationship means that modifying the side length by a scaling factor will change the volume by the cube of that factor.

Density

Density is defined as mass per unit volume. It quantifies how much mass is present in a specific volume and is affected by any change in the object's mass or volume, assuming the other variable remains constant.

Transcript

00:01 This problem deals with some algebraic manipulation of our density equation.

00:05 So to start off with, we are told we have a compressible cube with a density of row and a side length of l.

00:12 And we're saying that we keep the mass of this cube the same, but we are going to compress the length of each side to half of its original length.

00:21 So we want to know what is the new density in terms of the old density.

00:26 To do that, let's write what our old density is.

00:28 We know that the density of this cube will be equal to its mass divided by its volume.

00:33 And in this case, the volume of our cube is l times l times l, or we can write l cubed.

00:41 We know that it's l cubed because this is a cube.

00:44 The length, width, and height are all the same, and we can express that as l.

00:50 Now what we're going to say is we're going to deal with this new row, this new density.

00:55 I'm going to call it row prime.

00:57 I'm going to write it in red.

00:58 This new density.

01:01 The way we're going to get it is we're keeping mass the same, but instead of having the original side length, our side length is now half of what it used to be.

01:10 So instead of l times l times l it is 0 .5l times 0 .5l, which i can also write as 0 .125l cubed.

01:25 And what i can see right away is i know that original row, black row, is equal to m over l cubed.

01:32 In this expression, i have m over l cubed.

01:36 So instead of writing that, i can just say that row prime is equal to 1 over .125 times original row.

01:49 And instead of dealing with that fraction and a decimal, 1 divided by .125 is equal to 8.

01:56 So what we have now is our new density, row prime, is equal to eight times our old density.

02:03 And that makes sense because if we make our cubes smaller, its density should go up.

02:08 Now our second question says we're going to keep the same shape and mass, but we're asking, what would we need to do to l in order to result in three times the density? so again, we know that for this cube, density is equal to mass over l cubed, length times west times height is l times l times l...

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