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To understand how the surface area of a cube changes, let's start with the basics. The surface area of a cube is the total area of all six faces. Each face of a cube is a square, and if the side of the cube is denoted by \( s \), the area of one face is \( s^2 \). Since a cube has six faces, the formula for the surface area is \( 6s^2 \).

When the dimensions of a cube are changed, the surface area changes as well. In this exercise, the dimensions of a cube are tripled. This means the new side length of the cube is \( 3s \). To find the new surface area, substitute \( 3s \) into the surface area formula:
\[ 6(3s)^2 = 54s^2 \]
The result is that the surface area becomes nine times larger when the sides are tripled, as \( 54s^2 \) is nine times \( 6s^2 \).

The volume of a cube tells us how much space it occupies, and it is calculated by taking the cube of its side length. If \( s \) is the side length, then the volume is given by \( s^3 \).

Like surface area, the volume changes when the side length changes. In this exercise, each dimension is tripled, so the new side length is \( 3s \). The new volume is calculated by substituting \( 3s \) into the volume formula:
\[ (3s)^3 = 27s^3 \]
This tells us that the volume of the cube increases by a factor of 27 when each side is tripled. This is because \( 27s^3 \) is 27 times the original volume \( s^3 \).

Scaling dimensions refers to the process of changing the size of an object's dimensions by a certain factor. When all dimensions of a shape are multiplied by the same factor, we say that the shape is scaled. In the context of this exercise, the cube's dimensions are scaled by a factor of three.

This scaling affects the cube's properties significantly. For surface area, the scaling factor when dimensions change is the square of the scaling factor for the sides because the surface area is based on the square of the sides of the cube. As a result, if dimensions are tripled, the surface area increases by a factor of \( 3^2 = 9 \).

For volume, the scaling factor is the cube of the scaling factor for the sides because volume is based on the cube of the side length. So, tripling the side lengths results in the volume increasing by \( 3^3 = 27 \). These changes highlight the exponential nature of scaling in geometry.

Mathematical transformations involve changing geometric figures' size, position, or dimensions. In this exercise, the transformation applied to the cube is scaling, which affects both surface area and volume.

Transformations can be visualized as operations that modify geometric figures without altering their shape. Scaling, for example, maintains the shape of a figure while changing its size. This exercise focuses specifically on how a scale transformation of tripling dimensions impacts surface area and volume.

When considering transformations, it's crucial to understand:

• Scaling changes dimensions in a proportional way.
• Surface area and volume calculations are based on these modified dimensions.
• The impact is exponential: surface area scales with the square, and volume with the cube of the scaling factor.

Understanding these principles can also apply to different contexts, not just cubes, guiding us in geometry and design.