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Solve problems with proportions between similar bodies.
understand scale relationships
Tim has a cube with a side of 4 inches. It has a similar cube with dimensions twice the size of the first cube. like himVolumeof the larger cube relative to the volume of the smaller cube?
In this concept you will learn to understand the scale ratios ofAreaand volume
scale ratios
You can compare distance, area, and volume scale relationships when viewing three-dimensional figures. Some examples of three-dimensional shapes include a prism or a pyramid. When you compare different metrics, you see the proportional relationships between them.
Let's consider a situation where loudness is at stake.
Bach has onescale modelfrom a warehouse. A storage unit is shaped like a rectangular prism and has dimensions of 4 inches by 3 inches by 6 inches. If the model scale is \(0.5 \text{ inches}=2\text{ feet}\), what are the actual dimensions of the storage unit? What is the volume?
First, note that this problem has two parts. The first part is figuring out the actual dimensions since Brooke has a scale model. The second part is finding out the volume. Start with the scale to write a length, width, and height ratio.
\(\begin{aligned}
&\text {length} \quad \text {width} \quad \text {height}\\
&\frac{0{,}5\mathrm{pulgadas}}{2\mathrm{tortas}}=\frac{4\mathrm{pulgadas}}{x\mathrm{pulgadas}}\quad\frac{0{, }5\mathrm{pulgadas}} {2 \mathrm{tortas}}=\frac{3 \mathrm{pulgadas}}{x \mathrm{tortas}} \quad \frac{0{,}5 \mathrm{pulgadas} }{2 \mathrm{tortas}}=\frac {6 \mathrm{pulgadas}}{x \mathrm{tortas}}
\end{aligned}\)
Next, cross-multiply for each dimension.
\(\begin{matriz}{ccc}
\text {upper} & \text {ancho} & \text {alt} \\
\frac{0.5}{2}=\frac{4}{x} & \frac{0.5}{2}=\frac{3}{x} & \frac{0.5}{2}=\frac{6} {X} \\
0,5 x=2 \times 4 und 0,5 x=2 \times 3 und 0,5 x=2 \times 6 \\
0.5x=8 is 0.5x=6 is 0.5x=12
\end{matriz}\)
Then divide both sides by 0.5 to find x.
\(\begin{matriz}{ccc}
\text {upper} & \text {ancho} & \text {alt} \\
0.5x=8 and 0.5x=6 and 0.5x=12 \\
\frac{0.5 x}{0.5}=\frac{8}{0.5} & \frac{0.5 x}{0.5}=\frac{6}{0.5} & \frac{0.5 x}{0.5}=\frac {12}{0,5} \\
x=16 and x=12 and x=24
\end{matriz}\)
The answers are 16, 12 and 24.
The storage unit is 16 feet long, 12 feet wide, and 24 feet high.
Then you need to calculate the volume of the storage unit.
\(\begin{aligned} V&=l\times w\times h \\ V&=16\times 12\times 24 \\ V&=4608\end{aligned}\)
The answer is 4608.
The volume of the storage unit is \(4608\: ft^{3}\) or 4608 cubic feet.
There is a relationship between the base area of the prism and the volume of the prism. Let's look at how the area of the base of the prism is related to the volume of the prism using the storage unit problem.
\(\begin{aligned} A&=l\times w \\ A&=16\times 12 \\ A&=192\end{aligned}\)
The area of the prism is 192 square feet or 192 square feet.
Now to see the relationship between the volume of a prism and the area of a prism, divide the volume by the area.
\(\dfrac{V}{A}=4608192\)
\(\dfrac{V}{A}=24\)
The answer is 24.
Note that this is the height of the prism.
Example \(\PageIndex{1}\)
Earlier you had a problem with Tim's dice.
Tim has two cubes, with the larger one being twice the size of the smaller one. This means there is a scaling factor of 2.
Solution
First find the dimensions of the largest cube.
The smallest cube has a side length of 4 inches. Since this is a cube, the length = width = height = 4 inches.
The largest cube has a side length of 4 inches x 2 = 8 inches. So length = width = height = 8 inches.
Then find the volume of both cubes and compare.
\(\begin{aligned}
&\text { volume of the smallest cube } \quad \text { volume of the largest cube }\\
&\begin{matrix}{ll}
V=l \times w \times h & V=l \times w \times h \\
V=4 \mathrm{pulgadas} \times 4 \mathrm{pulgadas} \times 4 \mathrm{pulgadas} & V=8 \mathrm{pulgadas} \times 8 \mathrm{pulgadas} \times 8 \mathrm{pulgadas} \ \
V=64 \mathrm{pulgadas}^{3} & V=512 \mathrm{pulgadas}^{3}
\end{matriz}
\end{aligned}\)
Then compare the volume of the larger cube to that of the smaller cube.
\(\begin{aligned} \dfrac{\text{volume of the largest cube}}{\text{volume of the smallest cube}}&=\dfrac{512}{64} \\ \dfrac{\text{volume of the largest Cube's Cube}}{\text{Volume of the smallest cube}}&=8\end{aligned}\)
The answer is 8
The volume of the larger cube is 8 times the volume of the smaller cube.
Example \(\PageIndex{2}\)
Show that the height of the following prism can be found using the volume to area ratio for a prism 6 inches long, 5 inches wide, and 9 inches high.
Solution
First calculate the volume of the prism.
\(\begin{aligned} V&=l\times w\times h \\ V&=6\times 5\times 9 \\ V&=270\end{aligned}\)
Then calculate the area of the prism.
\(\begin{aligned} A&=l\vezes w \\ A&=6\vezes 5 \\ A&=30 \end{aligned}\)
Then divide the volume by the area.
\(\begin{aligned}\dfrac{V}{A}&=\dfrac{270}{30} \\\dfrac{V}{A}&=9\end{aligned}\)
The answer is 9
Note that this is the height of the prism.
Example \(\PageIndex{3}\)
A prism is 16 feet long, 12 feet wide, and 18 feet high. Find the volume of the prism.
Solution
\(\begin{aligned} V&=l\times w\times h \\ V&=16\times 12\times 18 \\ V&=3456\end{aligned}\)
The answer is 3456.
The volume of the prism is \(3456 ft^{3}\) or 3456 cubic feet.
Example \(\PageIndex{4}\)
A prism is 16 feet long, 12 feet wide, and 18 feet high. Find the ground plane.
Solution
\(\begin{aligned} A&=l\times w \\ A&=16\times 12 \\ A&=192\end{aligned}\)
The answer is 192.
The area of the prism is \(192 ft^{2}\) or 192 square feet.
Example \(\PageIndex{5}\)
Write a ratio comparing the volume to the area of the prism and simplify it.
Solution
\(\beign{aligned}\dfrac{V}{A}&=\dfrac{3456}{192} \\ \dfrac{V}{A}&=18\end{aligned}\)
The answer is 18.
The height of the prism is 18 feet.
Analyse
solve all problems
1. A cube measures 8 feet on a side. A similar cube has twice the dimensions. How is the volume of the larger cube related to the volume of the smaller cube? Write a reason for showing the comparison.
2. A cube measures 3 inches on each side. A similar cube has dimensions half the size of the other cube. How is the volume of the larger cube related to the volume of the smaller cube? Write a reason for showing the comparison.
3. A scale model of a sandbox has dimensions of 0.5 inch by 3 inch by 4 inch. If the model scale is 1 inch = 1 foot, what is the actual volume of the litter box?
4. A cube measures 5 inches on each side. A similar cube has three times larger dimensions. How is the volume of the larger cube related to the volume of the smaller cube? Write a reason for showing the comparison.
5. A shipping box measures 16 inches x 12 inches x 8 inches. A second box is similar in size, but each dimension is 14 times larger. How does the volume of the second box compare to the volume of the first box?
6. Rina's aquarium has a volume of 8,000 cubic inches. The dimensions of the AVA aquarium are 12 the size of Rina. What is the volume of the AVA aquarium?
7. A prism is 6 feet wide, 8 feet long, and 12 feet high. What is the volume of the prism?
8. What is the base of this prism?
9. What volume would a prism of the size described above have?
10. What is the volume of a prism of the size described above?
11. What volume would a prism twice as large as described above have?
12. What ratio can you use to find the ratio between volume and area?
13. What measure do you find when you simplify this ratio?
14. Right or wrong. You can use scale measurements to find the height of a prism.
15. Right or wrong. You can use scale measurements to find the dimensions of a prism.
resources
Review (Responses)
To view survey responses, open themPDF fileand look for Section 4.11.
vocabulary
| Expression | Definition |
|---|---|
| Area | Area is the space within the perimeter of a two-dimensional figure. |
| scale model | A scale model is a model that represents a three-dimensional space. |
| three dimensional | A three-dimensionally drawn figure is drawn using length, width, and height or depth. |
| two-dimensional | A figure drawn in two dimensions is drawn using only length and width. |
| Volume | Volume is the amount of space within the confines of a three-dimensional object. |
Video: Principles of Similar Solids - Basics
Activities: Area and Volume of Similar Solids Discussion Questions
Learning Aids: Study Guide for Surface Area and Volume
Exercise: Area and volume of similar bodies
Real World: Is There Life Out There?