To solve this problem, let's use the formula for the volume of a pyramid and take advantage of the information given:

Problem Recap:

• The pyramid has a height of $3$ meters.
• The base of the pyramid is a square with sides of $3$ meters.
• The cross-section of the pyramid perpendicular to the altitude at a distance $x$ meters down from the vertex is a square with sides $x$ meters.

We need to find the volume of the pyramid.

Approach:

The formula for the volume of a pyramid is:

$V = \frac{1}{3} \times \text{Base Area} \times \text{Height}$

In this case:

• The base area of the pyramid is $3 \times 3 = 9 \, \text{m}^2$.
• The height is given as $3 \, \text{m}$.

Thus, the volume of the pyramid is:

$V = \frac{1}{3} \times 9 \times 3 = \frac{1}{3} \times 27 = 9 \, \text{m}^3$

Explanation:

The cross-section description hints at a linear relationship between the side length of the square cross-section and the height from the vertex. However, since the question asks for the volume of the whole pyramid, we simply apply the standard formula for the volume of a pyramid with a square base.

Would you like further clarification on any steps?

Here are some related questions to extend your understanding:

1. How would the volume formula change if the base was not a square but a rectangle?
2. Can you derive the volume formula for a pyramid with any polygonal base?
3. What is the significance of the linear relationship between the cross-section and the height?
4. How would you compute the volume if the pyramid had a different height?
5. Can this method be applied to find the volume of a cone? If so, how?

Tip: For similar problems involving geometric shapes, visualizing the shape and its dimensions can significantly aid in understanding the relationships between different elements.