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Chapter 12: Problem 8

Two similar pyramids have volumes 3 and 375 . Find the ratios of: a. the heights b. the base areas c. the total areas

Short Answer

Expert verified
a. The ratio of the heights of the pyramids is \(1:5\). b. The ratio of the base areas of the pyramids is \(1:25\). c. The ratio of the total areas of the pyramids is \(1:25\).

Step by step solution

01

Find the Ratio of Heights

By use of formula, the ratio of the volumes of two similar figures is equal to the cube of the ratio of their corresponding lengths (or heights, in this context). So, if we are given that the volumes of two similar pyramids are \(3:375\) or \(3/375 = 1/125\), we can set up the formula: \((height1/height2)^3 = volume1/volume2\). Solving for \(height1/height2\), we find the cube root of \(1/125\), which gives us \(1/5\).
02

Find the Ratio of Base Areas

For similar figures, the ratio of their areas is the square of the ratio of their corresponding lengths. So the ratio of the base areas of the two pyramids will be \((height1/height2)^2\). Inserting our result from Step 1, \((1/5)^2 = 1/25\).
03

Find the Ratio of Total Areas

The same rule applies for total area as for base area, since they are both areas. So the ratio of the total areas of the two pyramids will also be \((height1/height2)^2\). Again, inserting our result from Step 1, \((1/5)^2 = 1/25\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Ratio of Volumes
In geometry, the concept of similarity can be powerful, especially when working with volumes of three-dimensional figures like pyramids. When you encounter similar pyramids, the ratio of their volumes can provide crucial insights.
  • Volume ratio of similar objects is determined by the cube of the ratio of their corresponding linear dimensions. This might include heights, widths, or lengths.
  • For example, consider two similar pyramids with volumes of 3 and 375. The volume ratio, therefore, is \(3/375\), which simplifies to \(1/125\).
This means for every unit of volume in the smaller pyramid, there are 125 units in the larger one.
Ratio of Heights
The height ratio between two similar geometric figures is derived from the cube root of the volume ratio. This might sound complex, but it's a method that directly links linear dimensions with volumetric properties.
  • To find the height ratio for pyramids with a volume ratio of \(1/125\), take the cube root of \(1/125\) to obtain \(1/5\).
  • This simplifies to mean the smaller pyramid is 1/5th the height of the larger pyramid.
Understanding this relationship is vital since it helps translate three-dimensional volumetric data back to one-dimensional linear data.
Ratio of Areas
When dealing with areas of similar figures, the square of the ratio of their linear dimensions (like height) becomes important. This is true for both base areas and total surface areas.
  • Given our height ratio from earlier of \(1/5\), the base area ratio becomes \((1/5)^2 = 1/25\).
  • This same logic applies to the total surface areas as well, making both the base and total area ratios equal to \(1/25\).
This highlights how linear dimensions powerfully influence two-dimensional properties like area.

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Most popular questions from this chapter

Copy and complete the table for spheres. $$\begin{array}{|l|c|}\hline \text { Radius } & \frac{3}{4} k \\ \hline \text { Area } & ? \\ \hline \text { Volume } & ? \\ \hline \end{array}$$

The purpose is to prove Theorem \(12-11\) for some similar solids. The bases of two similar right prisms are regular pentagons with base edges \(e_{1}\) and \(e_{2}\) and base areas \(B_{1}\) and \(B_{2} .\) The heights are \(h_{1}\) and \(h_{2} .\) Prove that the ratio of the lateral areas is \(e_{1}^{2}: e_{2}^{2}\)

Determine whether each conjecture is true or false. Give a counterexample for any false conjecture. \(\angle L\) and \(\angle M\) are complementary angles. \(\angle N\) and \(\angle P\) are complementary angles. If \(m \angle L=y-2, m \angle M=2 x+3, m \angle N=2 x-y,\) and \(m \angle P=x-1\) find the values of \(x, y, m \angle L, m \angle M, m \angle N,\) and \(m \angle P .\)

A diagonal of a cube joins two vertices not in the same face. If the diagonals are \(4 \sqrt{3} \mathrm{cm}\) long, what is the volume?

Find the volume of a regular hexagonal pyramid with height \(8 \mathrm{cm}\) and base edges \(6 \mathrm{cm}\). (GRAPH CANT COPY)
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