The Volumes Of Two Similar Solids Are 210 M 3 210 \, \text{m}^3 210 M 3 And 1 , 680 M 3 1,680 \, \text{m}^3 1 , 680 M 3 . The Surface Area Of The Larger Solid Is 856 M 2 856 \, \text{m}^2 856 M 2 . What Is The Surface Area Of The Smaller Solid?A. 107 M 2 107 \, \text{m}^2 107 M 2

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Introduction

Similar solids are three-dimensional shapes that have the same shape but not necessarily the same size. The ratio of the corresponding dimensions of similar solids is constant. In this article, we will explore the relationship between the volumes and surface areas of two similar solids. We will use the given information about the volumes and surface area of the larger solid to find the surface area of the smaller solid.

Understanding Similar Solids

Similar solids have the same shape but not necessarily the same size. The ratio of the corresponding dimensions of similar solids is constant. This means that if we have two similar solids, the ratio of their corresponding sides, heights, or radii is the same.

The Formula for the Volume of a Solid

The volume of a solid is given by the formula:

V = Bh

where V is the volume, B is the area of the base, and h is the height of the solid.

The Formula for the Surface Area of a Solid

The surface area of a solid is given by the formula:

SA = 2Bh + 2ph

where SA is the surface area, B is the area of the base, h is the height of the solid, and p is the perimeter of the base.

The Relationship Between the Volumes of Similar Solids

The ratio of the volumes of two similar solids is equal to the cube of the ratio of their corresponding dimensions. Mathematically, this can be expressed as:

V1 / V2 = (s1 / s2)^3

where V1 and V2 are the volumes of the two solids, and s1 and s2 are the corresponding dimensions.

The Relationship Between the Surface Areas of Similar Solids

The ratio of the surface areas of two similar solids is equal to the square of the ratio of their corresponding dimensions. Mathematically, this can be expressed as:

SA1 / SA2 = (s1 / s2)^2

where SA1 and SA2 are the surface areas of the two solids, and s1 and s2 are the corresponding dimensions.

Given Information

We are given the following information:

  • The volume of the smaller solid is 210 m^3.
  • The volume of the larger solid is 1,680 m^3.
  • The surface area of the larger solid is 856 m^2.

Finding the Surface Area of the Smaller Solid

We can use the formula for the ratio of the volumes of similar solids to find the ratio of the corresponding dimensions of the two solids:

V1 / V2 = (s1 / s2)^3

Substituting the given values, we get:

210 / 1680 = (s1 / s2)^3

Simplifying, we get:

0.125 = (s1 / s2)^3

Taking the cube root of both sides, we get:

s1 / s2 = 0.5

This means that the ratio of the corresponding dimensions of the two solids is 0.5.

We can use the formula for the ratio of the surface areas of similar solids to find the surface area of the smaller solid:

SA1 / SA2 = (s1 / s2)^2

Substituting the given values, we get:

856 / SA2 = (0.5)^2

Simplifying, we get:

856 / SA2 = 0.25

Multiplying both sides by SA2, we get:

214.4 = SA2

Therefore, the surface area of the smaller solid is approximately 214.4 m^2.

Conclusion

In this article, we explored the relationship between the volumes and surface areas of two similar solids. We used the given information about the volumes and surface area of the larger solid to find the surface area of the smaller solid. We found that the surface area of the smaller solid is approximately 214.4 m^2.

References

  • [1] "Similar Solids" by Math Open Reference
  • [2] "Volume and Surface Area of Solids" by Khan Academy

Discussion

What do you think about the relationship between the volumes and surface areas of similar solids? Do you have any questions or comments about this article? Please feel free to share your thoughts in the discussion section below.

Related Articles

  • [1] "The Volume and Surface Area of a Cube"
  • [2] "The Volume and Surface Area of a Sphere"
  • [3] "The Volume and Surface Area of a Cylinder"

Categories

  • Mathematics
  • Geometry
  • Similar Solids
  • Volume and Surface Area
    The volumes of two similar solids and the surface area of the larger solid: Q&A ====================================================================

Introduction

In our previous article, we explored the relationship between the volumes and surface areas of two similar solids. We used the given information about the volumes and surface area of the larger solid to find the surface area of the smaller solid. In this article, we will answer some frequently asked questions about similar solids and their volumes and surface areas.

Q: What is the definition of similar solids?

A: Similar solids are three-dimensional shapes that have the same shape but not necessarily the same size. The ratio of the corresponding dimensions of similar solids is constant.

Q: How do you find the ratio of the volumes of similar solids?

A: The ratio of the volumes of two similar solids is equal to the cube of the ratio of their corresponding dimensions. Mathematically, this can be expressed as:

V1 / V2 = (s1 / s2)^3

where V1 and V2 are the volumes of the two solids, and s1 and s2 are the corresponding dimensions.

Q: How do you find the ratio of the surface areas of similar solids?

A: The ratio of the surface areas of two similar solids is equal to the square of the ratio of their corresponding dimensions. Mathematically, this can be expressed as:

SA1 / SA2 = (s1 / s2)^2

where SA1 and SA2 are the surface areas of the two solids, and s1 and s2 are the corresponding dimensions.

Q: What is the relationship between the volumes and surface areas of similar solids?

A: The volumes and surface areas of similar solids are related by the following formulas:

V1 / V2 = (s1 / s2)^3 SA1 / SA2 = (s1 / s2)^2

Q: How do you find the surface area of the smaller solid if you know the surface area of the larger solid?

A: To find the surface area of the smaller solid, you can use the formula:

SA1 / SA2 = (s1 / s2)^2

Substitute the given values and solve for SA2.

Q: What is the surface area of the smaller solid if the surface area of the larger solid is 856 m^2 and the ratio of the corresponding dimensions is 0.5?

A: To find the surface area of the smaller solid, you can use the formula:

SA1 / SA2 = (s1 / s2)^2

Substitute the given values:

856 / SA2 = (0.5)^2

Simplifying, you get:

856 / SA2 = 0.25

Multiplying both sides by SA2, you get:

214.4 = SA2

Therefore, the surface area of the smaller solid is approximately 214.4 m^2.

Q: What are some real-world applications of similar solids?

A: Similar solids have many real-world applications, including:

  • Architecture: Similar solids are used in the design of buildings and bridges.
  • Engineering: Similar solids are used in the design of machines and mechanisms.
  • Physics: Similar solids are used to model the behavior of particles and systems.

Conclusion

In this article, we answered some frequently asked questions about similar solids and their volumes and surface areas. We hope that this article has been helpful in understanding the relationship between the volumes and surface areas of similar solids.

References

  • [1] "Similar Solids" by Math Open Reference
  • [2] "Volume and Surface Area of Solids" by Khan Academy

Discussion

Do you have any questions or comments about similar solids and their volumes and surface areas? Please feel free to share your thoughts in the discussion section below.

Related Articles

  • [1] "The Volume and Surface Area of a Cube"
  • [2] "The Volume and Surface Area of a Sphere"
  • [3] "The Volume and Surface Area of a Cylinder"

Categories

  • Mathematics
  • Geometry
  • Similar Solids
  • Volume and Surface Area