How does the volume of a cube change if the length of one side is increased by one unit, another side is decreased by one unit, and the third side is left unchanged? How does changing the shape of a container affect the volume of the matter?

The volume of a^3 (where a is a side length of the original cube) becomes

a*(a+1)(a-1) = a*(a^2 -1) = a^3 - a^2
, which is less than the original volume.

The fractional decrease in volume depends upon what the "units" of length are.

To understand how the volume of a cube changes when the length of one side is increased, another side is decreased, and the third side remains unchanged, let's break it down step by step.

1. Let's start with a cube whose side length is denoted by "x". The volume of a cube is given by V = x^3, where V represents the volume.

2. Now, if we increase the length of one side by one unit, the new side length can be represented as (x + 1).

3. Similarly, if we decrease the length of another side by one unit, the new side length can be represented as (x - 1).

4. The third side remains unchanged, so its length remains at "x".

5. The new volume of the cube, denoted as V', can be calculated using the new side lengths: V' = (x + 1)(x - 1)(x).

6. Expanding this equation, we get V' = (x^2 - 1)(x) = x^3 - x.

Comparing the new volume V' = x^3 - x with the original volume V = x^3, we see that the volume of the cube changes. In this case, it decreases by one unit, as indicated by the "-x" term in the equation.

Regarding the second part of your question about how changing the shape of a container affects the volume of matter, it primarily depends on the container's shape and its dimensions.

1. If you have a regular shape, such as a cube or rectangular prism, changing the shape will alter the volume according to the new dimensions. For example, if you change a rectangular prism into a cube by making all sides equal, the volume will change to a value that represents a cube rather than a rectangular prism.

2. If you have an irregular shape, such as a curved surface or an irregularly shaped container, changing the shape can have varying effects on the volume of matter. In this case, determining the volume might involve more complex calculations or measuring techniques.

In summary, changing the shape of a regular container will affect the volume based on the new dimensions, while with irregular containers, determining the volume can be more challenging and may require additional calculations or measurements.