Question

In what ratio should a plane parallel to the base of a pyrafnd divide its altitude so that the volumes of the parts into which the plane divides the pyramid have the ratio \(m: n\) ?

Short answer

The ratio is \( \left(\frac{m}{m+n}\right)^{1/3}:1-\left(\frac{m}{m+n}\right)^{1/3} \).

Step-by-step solution

Understanding the Problem

We need to find the ratio in which the altitude of a pyramid must be divided by a plane parallel to its base so that the volumes of the two resulting parts have a specific volume ratio, \( m:n \).

Setting Up the Volume Formula

The volume \( V \) of a pyramid is given by \( V = \frac{1}{3}Bh \), where \( B \) is the area of the base and \( h \) is the height (or altitude). When a plane cuts a pyramid parallel to its base, similar pyramids are formed.

Using Similarity of Pyramids

The key to solving this exercise is understanding that when a plane cuts a pyramid parallel to its base, the resulting smaller pyramid is similar to the original. The dimensions of the smaller pyramid are scaled by a factor \( k \), where \( k \) is the ratio of the smaller altitude to the full altitude \( h \).

Expressing Volume Ratio

Since the pyramids are similar, the smaller pyramid's volume \( V_s \) is \( V_s = \frac{1}{3}B(kh) = k^3 V \) because volume scales with the cube of the linear dimensions. This means the volume of the smaller pyramid is \( k^3 \) times the volume of the original pyramid.

Setting Up the Equation for Volume Ratio

Let the section at height \( kh \) from the base divide the volume of the pyramid such that the remaining volume is \( n \) times smaller than \( m \). The volume of the remaining frustum is \( V_r = V - V_s = V(1 - k^3) \). Therefore, the volume ratio of the smaller pyramid to the remaining frustum is \( \frac{k^3 V}{(1 - k^3)V} = \frac{m}{n} \).

Solving for the Height Ratio

From the volume ratio equation, we have \( \frac{k^3}{1 - k^3} = \frac{m}{n} \). Cross-multiply to get \( nk^3 = m - mk^3 \). Combine terms to get \( k^3(m + n) = m \). Solving for \( k^3 \) gives \( k^3 = \frac{m}{m+n} \). Thus, \( k = \left(\frac{m}{m+n}\right)^{1/3} \).

Conclusion

The height (altitude) of the pyramid must be divided in the ratio \( k:1-k \), where \( k = \left(\frac{m}{m+n}\right)^{1/3} \). This gives the desired volume ratio \( m:n \).

Key concepts

Pyramid Volume Ratio

When dealing with pyramids, understanding how their volume can be divided is crucial, especially when a plane parallel to the base cuts through it. If you want to divide a pyramid's volume into two parts with a specific ratio, say \( m:n \), it involves an interesting geometric exercise involving division of its altitude.
The volume \( V \) of a pyramid is expressed by the formula \( V = \frac{1}{3}Bh \), where \( B \) is the base area and \( h \) the altitude. When a plane parallel to the base divides the pyramid, the part of the pyramid above the plane is itself a pyramid, similar to the original but differently scaled.
The challenge lies in finding the exact point along the altitude where this division creates two volumes with the specified ratio \( m:n \). The division of volumes is closely tied to the ratio of the altitudes since volume scales as the cube of the linear dimensions in similar shapes.

Similar Pyramids

When a plane cuts through a pyramid parallel to its base, the resulting smaller pyramid is similar to the original pyramid. This means they share the same shape, but have different sizes. The smaller pyramid's dimensions are proportional to the larger pyramid's dimensions by a constant factor, known as the scale factor \( k \).
Due to this similarity, their linear dimensions scale by \( k \), but their volumes scale by \( k^3 \). This scaling relationship helps us understand how a plane's position in cutting the pyramid affects the division of the pyramid's total volume. This is crucial for solving problems that involve the division of volumes by altitude, such as when trying to achieve a specific volume ratio between two parts of a divided pyramid.
Understanding this concept allows one to set up the necessary calculations to determine the required division point on the pyramid’s altitude, ensuring the volumes meet the desired ratio.

Altitude Division

Dividing an altitude of a pyramid can be thought of as finding a specific point along this altitude where the cut should be made to achieve desired volume proportions. For a resulting volume that aligns with a given \( m:n \) ratio, the altitude must be precisely divided.
By considering the scale factor \( k \) for the similar pyramids, an equation is established: \( \frac{k^3}{1 - k^3} = \frac{m}{n} \). Solving this equation reveals that the altitude should be divided in the ratio \( k:(1-k) \), where \( k = \left(\frac{m}{m+n}\right)^{1/3} \).
This calculation reflects how far up from the original pyramid's base the plane should intersect the altitude to yield the exact volume ratios in the resulting pyramid and the frustum beneath it.

Plane Parallel to Base

A plane that is parallel to the base of a pyramid plays a special and significant role in geometric calculations. When it slices through the pyramid, it doesn’t just change the shape or size arbitrarily; it creates a smaller pyramidal section that maintains its geometric similarity with the original.
This plane can be positioned at various points along the altitude, effectively shifting the scale factor \( k \) and determining how the volume is divided within the pyramid. The symmetry and similar geometry created by such a plane enable a straightforward calculation regarding how the full pyramid's volume is allocated between the new smaller pyramid above the plane and the frustum that remains below it.
Thus, knowing where to place this plane is crucial for problems requiring a specific volume division, as it directly affects how the volume scales down alongside the altitude.