Sharon Is Drawing Up A Plan For Building A Reduced Eiffel Tower On Her Backyard Patio. She Has Room To Make A Base Of 5 Feet. Her Work Is Shown Below:France: Height = 1,050 Ft; Base = 328 Ft1. Proportion: $\[\frac{1050}{328}=\frac{5}{x}\\]2.

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Introduction

Imagine having a miniature version of the iconic Eiffel Tower in your own backyard. Sharon is on a mission to make this dream a reality by building a reduced-scale replica of the Eiffel Tower on her patio. However, she faces a challenge: she only has a base of 5 feet to work with. In this article, we will explore the mathematical concept of proportion and scaling to help Sharon determine the height of her miniature Eiffel Tower.

Understanding the Problem

The original Eiffel Tower in France stands at a height of 1,050 feet and has a base diameter of 328 feet. Sharon wants to create a reduced-scale model of the tower, but she only has a base of 5 feet to work with. To determine the height of her miniature tower, she needs to use the concept of proportion.

Setting Up the Proportion

A proportion is a statement that two ratios are equal. In this case, the ratio of the height of the original Eiffel Tower to its base is equal to the ratio of the height of Sharon's miniature tower to its base. We can set up the proportion as follows:

1050328=5x{\frac{1050}{328}=\frac{5}{x}}

where xx is the unknown height of Sharon's miniature tower.

Solving the Proportion

To solve the proportion, we can cross-multiply and then divide both sides by the coefficient of xx. This will give us the value of xx, which represents the height of Sharon's miniature tower.

1050328=5x{\frac{1050}{328}=\frac{5}{x}}

Cross-multiplying:

1050x=328×51050x = 328 \times 5

Expanding the right-hand side:

1050x=16401050x = 1640

Dividing both sides by 1050:

x=16401050x = \frac{1640}{1050}

Simplifying the fraction:

x=1.5647x = 1.5647

Rounding to two decimal places:

x=1.56x = 1.56

Conclusion

In this article, we used the concept of proportion to help Sharon determine the height of her miniature Eiffel Tower. By setting up a proportion and solving for the unknown height, we were able to find the value of xx, which represents the height of Sharon's miniature tower. The result is a height of approximately 1.56 feet, which is a scaled-down version of the original Eiffel Tower.

Real-World Applications

The concept of proportion is used in many real-world applications, including architecture, engineering, and design. By understanding how to set up and solve proportions, individuals can create scaled-down models of buildings, bridges, and other structures. This can be particularly useful in fields such as urban planning, where architects and engineers need to design buildings and infrastructure that are proportional to the surrounding environment.

Tips and Variations

  • To create a more accurate model of the Eiffel Tower, Sharon could use a 3D printer or other technology to create a more detailed and scaled-down version of the tower.
  • The concept of proportion can be applied to other structures, such as the Statue of Liberty or the Golden Gate Bridge.
  • In addition to architecture and engineering, the concept of proportion is also used in art and design, where artists and designers use proportions to create balanced and aesthetically pleasing compositions.

Mathematical Concepts

  • Proportion: a statement that two ratios are equal.
  • Ratio: a comparison of two quantities.
  • Scaling: the process of reducing or enlarging a quantity or object.
  • Similarity: the property of two or more objects being proportional to each other.

Glossary

  • Proportion: a statement that two ratios are equal.
  • Ratio: a comparison of two quantities.
  • Scaling: the process of reducing or enlarging a quantity or object.
  • Similarity: the property of two or more objects being proportional to each other.

References

  • [1] "The Eiffel Tower." Wikipedia, Wikimedia Foundation, 2023.
  • [2] "Proportion." Math Open Reference, 2023.
  • [3] "Scaling." Math Is Fun, 2023.
    Frequently Asked Questions: Scaling Down the Eiffel Tower ===========================================================

Q: What is the main concept used to solve the problem of scaling down the Eiffel Tower?

A: The main concept used to solve the problem of scaling down the Eiffel Tower is proportion. Proportion is a statement that two ratios are equal. In this case, the ratio of the height of the original Eiffel Tower to its base is equal to the ratio of the height of Sharon's miniature tower to its base.

Q: How do you set up a proportion?

A: To set up a proportion, you need to identify the two ratios that are equal. In this case, the two ratios are:

1050328=5x{\frac{1050}{328}=\frac{5}{x}}

where xx is the unknown height of Sharon's miniature tower.

Q: What is the formula for solving a proportion?

A: The formula for solving a proportion is:

ab=cd{\frac{a}{b}=\frac{c}{d}}

where aa and bb are the two quantities in the first ratio, and cc and dd are the two quantities in the second ratio.

Q: How do you solve a proportion?

A: To solve a proportion, you need to cross-multiply and then divide both sides by the coefficient of xx. This will give you the value of xx, which represents the height of Sharon's miniature tower.

Q: What is the importance of proportion in real-world applications?

A: Proportion is an important concept in many real-world applications, including architecture, engineering, and design. By understanding how to set up and solve proportions, individuals can create scaled-down models of buildings, bridges, and other structures.

Q: Can proportion be used in other areas of mathematics?

A: Yes, proportion can be used in other areas of mathematics, such as algebra and geometry. For example, proportion can be used to solve equations and inequalities, and to find the perimeter and area of shapes.

Q: What are some common mistakes to avoid when solving proportions?

A: Some common mistakes to avoid when solving proportions include:

  • Not cross-multiplying
  • Not dividing both sides by the coefficient of xx
  • Not checking the units of the quantities

Q: How can I apply proportion to real-world problems?

A: To apply proportion to real-world problems, you need to identify the two ratios that are equal and then set up a proportion. You can then solve the proportion using the formula and check your answer to make sure it is reasonable.

Q: What are some examples of real-world applications of proportion?

A: Some examples of real-world applications of proportion include:

  • Architecture: designing buildings and bridges
  • Engineering: designing machines and mechanisms
  • Design: creating scaled-down models of products and buildings

Q: Can proportion be used to solve problems in other areas of science?

A: Yes, proportion can be used to solve problems in other areas of science, such as physics and biology. For example, proportion can be used to solve problems involving motion and forces, and to analyze data in biology.

Q: What are some common misconceptions about proportion?

A: Some common misconceptions about proportion include:

  • Thinking that proportion is only used in mathematics
  • Thinking that proportion is only used in specific areas of mathematics
  • Thinking that proportion is a difficult concept to understand

Q: How can I practice solving proportions?

A: To practice solving proportions, you can try solving problems involving proportions, such as the one presented in this article. You can also try creating your own problems involving proportions and solving them.