Final answer:
To validate the relationship V = V × k^3 for a pyramid, you start with the smallest pyramid and calculate its volume using V = (1/3) × base area × height. After scaling the pyramid by a factor of k, the volume is scaled cubically, proving the relationship holds true.
Step-by-step explanation:
The student's question involves creating a pyramid with the smallest possible dimensions and then scaling it up to verify that the volume scales cubically, in other words, to verify that the relationship V = V × k^3 holds true. To answer this question, we need to get the volume of a pyramid, which can be calculated by the formula V = (1/3) × base area × height. Base length, base width, and height are linear dimensions that help us find these areas and volumes.
First, let's define the base of the smallest possible pyramid in terms of Pi (π). Let's say our pyramid has a square base for simplicity. If we choose the base length to be 2π and base width to be 2π, and the height to be π, the formula for volume becomes:
V = (1/3)×(2π)·(2π)·(π) = (4/3)π^3.
Next, if we scale the pyramid by a scale factor k, each of the linear dimensions (base length, base width, and height) gets multiplied by k. Thus, the new volume V' after scaling is:
V' = (1/3)×(k×2π)·(k×2π)·(k×π) = (1/3)×(2kπ)·(2kπ)·(kπ) = (4/3)k^3π^3.
This result shows that the volume indeed scales as the cube of the scale factor, verifying that V = V × k^3. When we apply the scale factor, each dimension increases proportionally, and since volume involves three dimensions (length, width, height), the volume scales by the cube of the scale factor.