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The volume of a cube, which is a three-dimensional square, depends on the length of its sides. The formula for calculating the volume of a cube is given by: \[ V = s^3 \] where

• \( V \) represents the volume
• \( s \) is the length of one side of the cube

Let's explore what happens when you double the side length of the cube. If the original side length is \( s \), doubling it means the new side length becomes \( 2s \). Now, plug the new side length into the volume formula: \[ V_{\text{new}} = (2s)^3 = 8s^3 \] As you can see, the new volume is eight times larger than the original volume. This happens because the volume of a cube changes with the cube of the side length. So, when each side is doubled, the volume increases by a factor of eight. Whenever you change the side length of a cube, remember this cube relationship for volume scaling.

The volume of a sphere, which is a perfectly round three-dimensional object, is determined by its radius. The formula for calculating the volume of a sphere is: \[ V = \frac{4}{3}\pi r^3 \] where

• \( V \) represents the volume
• \( r \) is the radius of the sphere

Now, let's see what happens when you double the radius of the sphere. If the original radius is \( r \), then doubling it means the new radius becomes \( 2r \). Plug the new radius into the volume formula: \[ V_{\text{new}} = \frac{4}{3}\pi (2r)^3 = \frac{4}{3}\pi 8r^3 = 8 \frac{4}{3}\pi r^3 \] Just like the cube, the new volume of the sphere is also eight times larger than the original volume. This is because the volume of a sphere changes with the cube of its radius. Therefore, doubling the radius results in an eightfold increase in volume. This cube relationship for volume applies to spheres as well.

Geometric transformations involve changing the size, position, or shape of a geometric figure. When dealing with the volume of shapes such as cubes and spheres, scaling transformations are particularly important. Scaling transformations involve enlarging or reducing a figure proportionally. For example:

• **Scaling a cube** means adjusting its side length proportionally. Doubling each side length results in an eightfold increase in volume because volume scales with the cube of the side length.
• **Scaling a sphere** involves changing its radius. Doubling the radius results in an eightfold increase in volume, again scaling with the cube of the radius.
• **General scaling laws**: For any three-dimensional shape, scaling the dimensions by a factor \( k \) results in the volume changing by a factor \( k^3 \).

Understanding these principles of geometric transformations helps to anticipate how the volume of different shapes will change under scaling, making it easier to solve problems related to resizing in geometry.