Select The Correct Answer.Cube $A$ And Cube $B$ Are Similar Solids. The Volume Of Cube $A$ Is 27 Cubic Inches, And The Volume Of Cube $B$ Is 125 Cubic Inches. How Many Times Larger Is The Base Area Of Cube
Introduction
In geometry, similar solids are three-dimensional shapes that have the same shape but not necessarily the same size. When two similar solids have the same scale factor, their corresponding dimensions, such as side lengths and areas, are proportional. In this article, we will explore the concept of similar solids and how to determine the ratio of their base areas using their volumes.
Volume of Similar Solids
The volume of a cube is given by the formula V = s^3, where s is the length of a side. Since cube A and cube B are similar, their corresponding side lengths are proportional. Let's denote the side length of cube A as s_A and the side length of cube B as s_B. We can write the ratio of their volumes as:
V_A / V_B = (s_A)^3 / (s_B)^3
Given that the volume of cube A is 27 cubic inches and the volume of cube B is 125 cubic inches, we can substitute these values into the equation:
27 / 125 = (s_A)^3 / (s_B)^3
Finding the Scale Factor
To find the scale factor between the two cubes, we need to take the cube root of both sides of the equation:
(s_A / s_B) = ∛(27 / 125)
Using a calculator, we can evaluate the cube root:
(s_A / s_B) ≈ 0.6
This means that the side length of cube A is approximately 60% of the side length of cube B.
Base Area Ratio
The base area of a cube is given by the formula A = s^2, where s is the length of a side. Since the side lengths of cube A and cube B are proportional, their base areas are also proportional. Let's denote the base area of cube A as A_A and the base area of cube B as A_B. We can write the ratio of their base areas as:
A_A / A_B = (s_A)^2 / (s_B)^2
Using the scale factor we found earlier, we can substitute (s_A / s_B) ≈ 0.6 into the equation:
A_A / A_B = (0.6)^2
Evaluating the expression, we get:
A_A / A_B ≈ 0.36
This means that the base area of cube A is approximately 36% of the base area of cube B.
Conclusion
In this article, we explored the concept of similar solids and how to determine the ratio of their base areas using their volumes. We found that the base area of cube A is approximately 36% of the base area of cube B. This demonstrates the importance of understanding the relationships between the dimensions of similar solids and how to apply this knowledge to solve problems in geometry.
Real-World Applications
The concept of similar solids has numerous real-world applications in fields such as engineering, architecture, and design. For example, when designing a building, architects need to consider the scale factor between different components, such as the ratio of the base area of the foundation to the base area of the roof. By understanding the relationships between the dimensions of similar solids, architects can create more efficient and aesthetically pleasing designs.
Common Mistakes to Avoid
When working with similar solids, it's essential to avoid common mistakes such as:
- Assuming that the scale factor between two similar solids is always 1:1.
- Failing to consider the relationships between the dimensions of similar solids.
- Not using the correct formulas to calculate the volume and base area of similar solids.
By avoiding these mistakes and understanding the concept of similar solids, you can develop a deeper appreciation for the relationships between the dimensions of three-dimensional shapes and apply this knowledge to solve problems in geometry.
Practice Problems
- Two similar cubes have volumes of 64 cubic inches and 216 cubic inches. What is the ratio of their base areas?
- A cube with a volume of 125 cubic inches is similar to a cube with a volume of 27 cubic inches. What is the ratio of their side lengths?
- A building has a base area of 100 square feet and a height of 50 feet. If the building is similar to a smaller building with a base area of 25 square feet and a height of 25 feet, what is the ratio of their base areas?
Solutions
- Using the formula V = s^3, we can find the ratio of their side lengths:
(64 / 216) = (s_A / s_B)^3
Taking the cube root of both sides, we get:
(s_A / s_B) ≈ 0.7
The ratio of their base areas is:
A_A / A_B = (s_A)^2 / (s_B)^2 ≈ (0.7)^2 ≈ 0.49
- Using the formula V = s^3, we can find the ratio of their side lengths:
(125 / 27) = (s_A / s_B)^3
Taking the cube root of both sides, we get:
(s_A / s_B) ≈ 1.6
- Using the formula A = s^2, we can find the ratio of their base areas:
A_A / A_B = (s_A)^2 / (s_B)^2
Since the buildings are similar, their corresponding dimensions are proportional. Let's denote the side length of the smaller building as s_B. We can write the ratio of their side lengths as:
(s_A / s_B) = ∛(100 / 25)
Evaluating the expression, we get:
(s_A / s_B) ≈ 2
The ratio of their base areas is:
Q: What is the difference between similar solids and congruent solids?
A: Similar solids are three-dimensional shapes that have the same shape but not necessarily the same size. Congruent solids, on the other hand, are three-dimensional shapes that have the same size and shape.
Q: How do you determine if two solids are similar?
A: To determine if two solids are similar, you need to check if their corresponding dimensions, such as side lengths and areas, are proportional. You can use the formula V = s^3 to find the ratio of their volumes and then take the cube root to find the ratio of their side lengths.
Q: What is the scale factor between two similar solids?
A: The scale factor between two similar solids is the ratio of their corresponding dimensions, such as side lengths and areas. It is usually denoted by the letter k.
Q: How do you find the scale factor between two similar solids?
A: To find the scale factor between two similar solids, you need to find the ratio of their volumes and then take the cube root. For example, if the volume of cube A is 27 cubic inches and the volume of cube B is 125 cubic inches, you can find the scale factor as follows:
(s_A / s_B) = ∛(27 / 125)
Q: What is the relationship between the volume and base area of similar solids?
A: The volume and base area of similar solids are related by the formula V = k^3 * A, where k is the scale factor. This means that the volume of a similar solid is equal to the base area of the original solid multiplied by the cube of the scale factor.
Q: How do you find the ratio of the base areas of two similar solids?
A: To find the ratio of the base areas of two similar solids, you need to find the ratio of their side lengths and then square it. For example, if the side length of cube A is 3 inches and the side length of cube B is 6 inches, you can find the ratio of their base areas as follows:
A_A / A_B = (s_A)^2 / (s_B)^2 = (3)^2 / (6)^2 = 9 / 36 = 1 / 4
Q: What is the significance of the scale factor in similar solids?
A: The scale factor is a critical concept in similar solids because it allows us to compare the dimensions of two similar solids. It is used to find the ratio of their volumes, base areas, and other dimensions.
Q: Can you give an example of how the scale factor is used in real-world applications?
A: Yes, the scale factor is used in many real-world applications, such as:
- Architecture: Architects use the scale factor to design buildings and other structures that are similar in shape but different in size.
- Engineering: Engineers use the scale factor to design machines and other devices that are similar in shape but different in size.
- Design: Designers use the scale factor to create prototypes and models of products that are similar in shape but different in size.
Q: What are some common mistakes to avoid when working with similar solids?
A: Some common mistakes to avoid when working with similar solids include:
- Assuming that the scale factor between two similar solids is always 1:1.
- Failing to consider the relationships between the dimensions of similar solids.
- Not using the correct formulas to calculate the volume and base area of similar solids.
Q: How can you practice working with similar solids?
A: You can practice working with similar solids by:
- Solving problems that involve similar solids.
- Creating your own examples of similar solids.
- Using online resources and tools to visualize and calculate the dimensions of similar solids.
Q: What are some additional resources for learning about similar solids?
A: Some additional resources for learning about similar solids include:
- Textbooks and online courses that cover geometry and similar solids.
- Online resources and tools that provide interactive examples and exercises.
- Professional organizations and communities that provide resources and support for learning about similar solids.