Two Similar Solids Have A Scale Factor Of 6 : 7 6:7 6 : 7 . What Is The Ratio Of Their Volumes Expressed In Lowest Terms?

When dealing with similar solids, it's essential to understand the relationship between their scale factors and volume ratios. In this article, we'll explore how to calculate the ratio of volumes for two similar solids with a given scale factor.

What is a Scale Factor?

A scale factor is a ratio that describes the proportional relationship between the corresponding dimensions of two similar figures. In the case of two similar solids, the scale factor is the ratio of the lengths of their corresponding sides.

Given Scale Factor: 6:7

We're given that the scale factor between the two similar solids is 6:7. This means that the ratio of the corresponding sides of the two solids is 6:7.

Calculating the Volume Ratio

The volume of a solid is directly proportional to the cube of its linear dimensions. Therefore, if we know the scale factor between two similar solids, we can calculate the ratio of their volumes by cubing the scale factor.

Let's denote the scale factor as $k = \frac{6}{7}$. To find the ratio of the volumes, we need to cube the scale factor:

$$\left(\frac{6}{7}\right)^3 = \frac{216}{343}$$

Simplifying the Volume Ratio

The volume ratio we obtained is $\frac{216}{343}$. However, we need to express it in its lowest terms. To do this, we can find the greatest common divisor (GCD) of the numerator and denominator and divide both by the GCD.

The GCD of 216 and 343 is 1. Therefore, the volume ratio in its lowest terms is:

$$\frac{216}{343}$$

Conclusion

In conclusion, the ratio of the volumes of two similar solids with a scale factor of 6:7 is $\frac{216}{343}$. This means that the volume of the larger solid is $\frac{216}{343}$ times the volume of the smaller solid.

Real-World Applications

Understanding the relationship between scale factors and volume ratios has numerous real-world applications. For example, in architecture, engineers need to calculate the volume of similar structures, such as buildings or bridges, to ensure that they are proportional and meet the required specifications.

Example Problems

1. Two similar cubes have a scale factor of 3:4. What is the ratio of their volumes?
2. A scale model of a building has a scale factor of 1:50. What is the ratio of the volume of the model to the actual building?

Solution to Example Problems

1. To find the ratio of the volumes, we need to cube the scale factor:

$$\left(\frac{3}{4}\right)^3 = \frac{27}{64}$$

Therefore, the ratio of the volumes is $\frac{27}{64}$.

2. To find the ratio of the volume of the model to the actual building, we need to cube the scale factor:

$$\left(\frac{1}{50}\right)^3 = \frac{1}{125000}$$

Therefore, the ratio of the volume of the model to the actual building is $\frac{1}{125000}$.

Final Thoughts

In conclusion, understanding the relationship between scale factors and volume ratios is crucial in various fields, including mathematics, architecture, and engineering. By applying the concept of scale factors and volume ratios, we can calculate the volume of similar solids and ensure that they meet the required specifications.

References

• [1] "Similar Solids" by Math Open Reference
• [2] "Scale Factors and Volume Ratios" by Khan Academy

Glossary

• Scale Factor: A ratio that describes the proportional relationship between the corresponding dimensions of two similar figures.
• Volume Ratio: The ratio of the volumes of two similar solids.
• Greatest Common Divisor (GCD): The largest number that divides two or more numbers without leaving a remainder.
  Frequently Asked Questions (FAQs) About Scale Factors and Volume Ratios ====================================================================

In this article, we'll address some of the most common questions about scale factors and volume ratios. Whether you're a student, teacher, or professional, these FAQs will provide you with a better understanding of this important concept.

Q: What is a scale factor?

A: A scale factor is a ratio that describes the proportional relationship between the corresponding dimensions of two similar figures. In the case of two similar solids, the scale factor is the ratio of the lengths of their corresponding sides.

Q: How do I calculate the scale factor between two similar solids?

A: To calculate the scale factor, you need to know the lengths of the corresponding sides of the two solids. For example, if the length of one side of the smaller solid is 6 units and the length of the corresponding side of the larger solid is 7 units, the scale factor is 6:7.

Q: What is the relationship between scale factors and volume ratios?

A: The volume of a solid is directly proportional to the cube of its linear dimensions. Therefore, if we know the scale factor between two similar solids, we can calculate the ratio of their volumes by cubing the scale factor.

Q: How do I calculate the volume ratio between two similar solids?

A: To calculate the volume ratio, you need to cube the scale factor. For example, if the scale factor is 6:7, the volume ratio is $\left(\frac{6}{7}\right)^3 = \frac{216}{343}$.

Q: Can I simplify the volume ratio?

A: Yes, you can simplify the volume ratio by finding the greatest common divisor (GCD) of the numerator and denominator and dividing both by the GCD.

Q: What is the greatest common divisor (GCD)?

A: The greatest common divisor (GCD) is the largest number that divides two or more numbers without leaving a remainder.

Q: How do I find the GCD of two numbers?

A: There are several ways to find the GCD of two numbers. One common method is to list the factors of each number and find the largest factor they have in common.

Q: What are some real-world applications of scale factors and volume ratios?

A: Scale factors and volume ratios have numerous real-world applications, including architecture, engineering, and design. For example, architects use scale factors to ensure that buildings are proportional and meet the required specifications.

Q: Can I use scale factors and volume ratios to compare the volumes of different shapes?

A: Yes, you can use scale factors and volume ratios to compare the volumes of different shapes. For example, if you know the scale factor between two similar cubes, you can calculate the ratio of their volumes.

Q: What are some common mistakes to avoid when working with scale factors and volume ratios?

A: Some common mistakes to avoid when working with scale factors and volume ratios include:

• Not cubing the scale factor when calculating the volume ratio
• Not simplifying the volume ratio by finding the GCD
• Not using the correct scale factor when comparing the volumes of different shapes

Q: How can I practice working with scale factors and volume ratios?

A: You can practice working with scale factors and volume ratios by:

• Using online calculators or software to calculate scale factors and volume ratios
• Creating your own examples and problems to solve
• Working with real-world applications, such as architecture or engineering projects

Conclusion

In conclusion, scale factors and volume ratios are essential concepts in mathematics and have numerous real-world applications. By understanding the relationship between scale factors and volume ratios, you can calculate the volume of similar solids and ensure that they meet the required specifications.