https://file.aiquickdraw.com/mathbot/file/1728661636887a4jog26x.jpg

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.

A pyramid 3 m high has a square base that is 3 m on a side. The cross-section of the pyramid perpendicular to the altitude x m down from the vertex is a square x m on a side. Find the volume of the pyramid.

Learn how to find the volume of a pyramid with a square base that is 3 m on each side and a height of 3 m. This problem involves applying the formula for the volume of a pyramid and understanding geometric principles. Suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation