Question

A pyramid with the altitude \(h\) is divided by two planes parallel to the base into three parts whose volumes have the ratio \(l: m: n\). Find the distances of these planes from the vertex.

Short answer

The distances are \( h_1 = h \sqrt[3]{\frac{l}{l+m+n}} \) and \( h_2 = h \sqrt[3]{\frac{l+m}{l+m+n}} \).

Step-by-step solution

Understand the Problem

We have a pyramid with altitude \( h \) divided by two planes parallel to the base into parts with volume ratios \( l:m:n \). We need to find distances from the vertex to these planes.

Recall the Volume Formula

The volume \( V \) of any section of the pyramid is proportional to the cube of the height from the vertex to the base of that section. Mathematically, \( V \propto h^3 \).

Setup the Ratios

Since the ratios of the volumes are \( l: m: n \), we express the heights of the sections as \( h_1 \), \( h_2 \), and \( h \/ \), where the volumes are proportional to \( h_1^3 \), \( (h_2 - h_1)^3 \), and \( (h - h_2)^3 \) respectively.

Express Heights in Terms of Ratios

Since the volume ratio is \( l:m:n \) for \( V_1:V_2:V_3 \), we set up equations: \( \frac{h_1^3}{(h_2 - h_1)^3} = \frac{l}{m} \) and \( \frac{(h_2 - h_1)^3}{(h - h_2)^3} = \frac{m}{n} \).

Solve for \( h_1 \) and \( h_2 \)

Solving the equations involves expressing \( h_1 \) and \( h_2 - h_1 \) in terms of \( l, m, n , \) and \( h \). The first ratio gives \( h_1 = h \sqrt[3]{\frac{l}{l+m+n}} \), and for the second plane, \( h_2 = h \sqrt[3]{\frac{l+m}{l+m+n}} \).

Verify Solution

Check to ensure the derived expressions satisfy the volume ratio \( l: m:n \), confirming the correctness of the distances derived.

Key concepts

Pyramid Geometry

When we look at pyramid geometry, we deal with a three-dimensional solid having a base that can be any polygon and triangular faces that converge at a common point, called the apex or vertex. In mathematical terms, a pyramid with a polygonal base is defined by:

• The base length and shape.
• The slant heights of the triangular faces.
• The altitude, which is the perpendicular distance from the base to the apex.

The most common pyramids are square-based, like the famous pyramids of Egypt. However, in geometry problems, we often generalize to polygons with sides, mean that the formula and understanding must be adaptable.

To solve problems involving pyramids, especially those related to volume, one must be comfortable using formulas. The basic volume formula for any pyramid is:\[V = \frac{1}{3} \times \text{Base Area} \times \text{Height}.\]This formula arises because a pyramid can be visualized as a structure similar to cones in geometry but with a polygon base.

Volume Ratio

In geometry problems like the one presented, a volume ratio can help us understand how different sections of a pyramid compare to each other in terms of size. When a pyramid is sliced by planes parallel to its base, it creates smaller, similar pyramidal sections. The unique property here is:

• The volume of these smaller sections is directly related to the cube of their respective altitudinal heights from their tops to the base.

This means, if you have multiple sections divided into a ratio like \( l:m:n \), it signifies:

• The first section has a volume proportional to \( h_1^3 \),
• The middle section's volume is proportional to \( (h_2 - h_1)^3 \),
• The last section's volume is proportional to \( (h - h_2)^3 \).

The given exercise required establishing a relationship between these volumes by setting up equations using these proportions, which were critical to deducing the actual distances from the vertex to the planes.

Altitude of a Pyramid

The altitude of a pyramid is a critical element in determining the proportions of divided volumes. The full altitude \( h \) refers to the distance from the apex to the centroid of the base, typically taken as perpendicular.In volume slicing problems, like dividing a pyramid into parts by parallel planes, the concept of altitude is key. The derived sections have smaller altitudinal portions which contribute to distinct section volumes.

• The section from the vertex to the first parallel plane has an altitude \( h_1 \).
• The section between the two planes contributes an altitude difference \( h_2 - h_1 \).
• The final section up to the base encompasses an altitude \( h - h_2 \).

Understanding and manipulating these heights using the cube root allows us to express and calculate the required distance for each section, ensuring our solution to the volume ratios is both logically grounded and mathematically precise.