Question

Compute the volumes of two similar polyhedra, if their total volume is \(V\), and their homologous edges have the ratio \(m: n\).

Short answer

Volumes are \( \frac{m^3 V}{m^3+n^3} \) and \( \frac{n^3 V}{m^3+n^3} \).

Step-by-step solution

Understanding Similar Polyhedra

Two similar polyhedra have the same shape but differ in size. The ratio of their corresponding linear dimensions is given as \( m:n \). This implies all corresponding lengths, such as edges, have this ratio. Similar polyhedra will have their volumes in the ratio \((m^3 : n^3)\).

Setting Up Volume Ratios

Given that the ratio of corresponding edges is \( m:n \), the ratio of their volumes will be \( m^3:n^3 \). This means if the two polyhedra have volumes \( V_1 \) and \( V_2 \), then \( \frac{V_1}{V_2} = \frac{m^3}{n^3} \).

Solving for Individual Volumes

Let the volumes of the two polyhedra be \( V_1 \) and \( V_2 \) and their total volume is \( V \), so \( V_1 + V_2 = V \). Using the volume ratio, \( V_1 = \frac{m^3}{n^3}V_2 \). Substitute into the total volume equation: \( \frac{m^3}{n^3}V_2 + V_2 = V \). Simplify to find \( V_2 = \frac{V}{1 + \frac{m^3}{n^3}} \).

Calculating the First Volume

Rearrange the equation from Step 3 to solve for \( V_1 \): \( V_1 = \frac{m^3}{n^3} \times \frac{V}{1 + \frac{m^3}{n^3}} \). Simplify the expression: \( V_1 = \frac{m^3 V}{m^3 + n^3} \).

Calculate the Second Volume

Using \( V_2 = \frac{V}{1 + \frac{m^3}{n^3}} \), simplify to find \( V_2 = \frac{n^3 V}{m^3 + n^3} \). These calculations ensure the two volumes sum to the total \( V \).

Key concepts

Volume Calculation

To determine the volumes of similar polyhedra, understanding their relationship is essential. Similar polyhedra are geometrically the same but vary in scale. Their volumes depend on the ratio of their homologous edge lengths. Given the total volume of two similar polyhedra and the edge ratio, you can use basic algebraic equations to find the individual volumes of each polyhedron.

Here's the process:

• Identify the edge ratio, usually given as \( m:n \).
• Understand that the volume ratio will be \( m^3:n^3 \) because volumes are a function of cubic dimensions.
• Add the individual volumes to equal the total volume \( V \).
• Solve the equations set up based on these relationships to find each volume separately.

This approach allows you to calculate accurately through multiplication and division rather than directly measuring or approximating.

Volume Ratio

The volume ratio of similar polyhedra is derived from the ratios of their corresponding linear dimensions. When the ratio of similar polyhedra's edge lengths is \( m:n \), the volumes will have a ratio of \( m^3:n^3 \).

This is because the volume of any polyhedron is proportional to the cube of its dimensions. Think of it this way: Each dimension scaled by \( m \) or \( n \) gets cubed, yielding \( m^3 \) and \( n^3 \). Thus:

• If \( (m:n) = (3:2) \), then the volume ratio is \( (3^3:2^3) = (27:8) \).
• This ratio suggests how much larger one polyhedron’s volume is compared to the other.

Using this ratio, you can determine how volumes change with varying dimensions and how they are distributed across polyhedra sharing a total volume.

Linear Dimensions Ratio

The linear dimensions ratio defines the scale difference between two similar polyhedra. This is given as \( m:n \), where each corresponding edge of one polyhedron is \( m \) times the size of the corresponding edge on the other.

This ratio helps in understanding how polyhedra sizes compare:

• Edge Ratio: Simply compare each side length.
• This ratio simplifies finding volume changes as the change in any linear dimension is uniform.
• If one polyhedron is scaled up, the entire set of linear dimensions uses that scale factor.

Knowing this ratio allows you to relate all identical measurements of two similar structures, making calculations, such as volumes, straightforward.

Stereometry

Stereometry is the branch of geometry that deals with the study of solid figures in three-dimensional space. In relation to similar polyhedra, stereometry provides the tools to understand and compute spatial properties.

The significance of stereometry in this context:

• It provides formulas for volume calculation relevant to three-dimensional shapes.
• Understanding stereometry helps in visualizing how taking specific portions of space relates differently in similar structures.
• Critical for applications where understanding and manipulating 3D shapes is necessary, like architecture or engineering.

Stereometry bridges the gap between abstract mathematical computations and real-world 3D applications, making it easier to reason about the scale and proportion of different solid shapes.