Ricardo Has A Square Hot Tub. He Wants To Build A Square Pool Next To It That Is A Dilation Of The Hot Tub Using A Scale Factor Of 5.If The Pool Is To Be 24 Ft On Each Side, What Is The Length Of One Side Of The Hot Tub?A. 4 Ft B. 4.8 Ft C. 6 Ft D.

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Introduction

Dilation is a fundamental concept in mathematics that involves changing the size of a shape or a figure. It is an essential topic in geometry, and understanding dilation and scale factors is crucial for solving various mathematical problems. In this article, we will explore the concept of dilation and scale factors, and we will use a real-world example to demonstrate how to apply these concepts to solve a problem.

What is Dilation?

Dilation is a transformation that changes the size of a shape or a figure. It involves scaling the shape up or down by a certain factor, known as the scale factor. The scale factor is a ratio that compares the size of the original shape to the size of the transformed shape.

Understanding Scale Factors

A scale factor is a ratio that compares the size of the original shape to the size of the transformed shape. It is usually represented by a number or a fraction. For example, if a shape is dilated by a scale factor of 2, it means that the size of the shape is doubled.

Applying Dilation and Scale Factors to Real-World Problems

Now that we have a good understanding of dilation and scale factors, let's apply these concepts to a real-world problem. Ricardo has a square hot tub, and he wants to build a square pool next to it that is a dilation of the hot tub using a scale factor of 5. If the pool is to be 24 ft on each side, what is the length of one side of the hot tub?

Step 1: Understand the Problem

The problem states that Ricardo wants to build a square pool next to his square hot tub. The pool is to be a dilation of the hot tub using a scale factor of 5. This means that the size of the pool will be 5 times the size of the hot tub.

Step 2: Identify the Given Information

The given information is:

  • The pool is to be 24 ft on each side.
  • The scale factor is 5.

Step 3: Apply the Scale Factor

To find the length of one side of the hot tub, we need to apply the scale factor. Since the scale factor is 5, we can set up a proportion to relate the size of the pool to the size of the hot tub.

Let x be the length of one side of the hot tub. Then, the length of one side of the pool is 5x.

We know that the length of one side of the pool is 24 ft, so we can set up the following equation:

5x = 24

Step 4: Solve for x

To solve for x, we can divide both sides of the equation by 5:

x = 24/5

x = 4.8

Conclusion

In this article, we explored the concept of dilation and scale factors in mathematics. We used a real-world example to demonstrate how to apply these concepts to solve a problem. Ricardo has a square hot tub, and he wants to build a square pool next to it that is a dilation of the hot tub using a scale factor of 5. If the pool is to be 24 ft on each side, the length of one side of the hot tub is 4.8 ft.

Key Takeaways

  • Dilation is a transformation that changes the size of a shape or a figure.
  • A scale factor is a ratio that compares the size of the original shape to the size of the transformed shape.
  • To apply dilation and scale factors to real-world problems, we need to understand the given information, identify the scale factor, and apply it to the problem.

Practice Problems

  1. A square is dilated by a scale factor of 3. If the length of one side of the original square is 6 cm, what is the length of one side of the dilated square?
  2. A rectangle is dilated by a scale factor of 2. If the length of one side of the original rectangle is 8 cm, what is the length of one side of the dilated rectangle?
  3. A triangle is dilated by a scale factor of 4. If the length of one side of the original triangle is 12 cm, what is the length of one side of the dilated triangle?

Answer Key

  1. 18 cm
  2. 16 cm
  3. 48 cm
    Dilation and Scale Factors: A Q&A Guide =============================================

Introduction

Dilation and scale factors are fundamental concepts in mathematics that involve changing the size of a shape or a figure. In our previous article, we explored the concept of dilation and scale factors and applied them to a real-world problem. In this article, we will provide a Q&A guide to help you understand dilation and scale factors better.

Q: What is dilation?

A: Dilation is a transformation that changes the size of a shape or a figure. It involves scaling the shape up or down by a certain factor, known as the scale factor.

Q: What is a scale factor?

A: A scale factor is a ratio that compares the size of the original shape to the size of the transformed shape. It is usually represented by a number or a fraction.

Q: How do I apply dilation and scale factors to real-world problems?

A: To apply dilation and scale factors to real-world problems, you need to understand the given information, identify the scale factor, and apply it to the problem. You can use the following steps:

  1. Understand the problem: Read the problem carefully and identify the given information.
  2. Identify the scale factor: Determine the scale factor that is being applied to the shape.
  3. Apply the scale factor: Use the scale factor to find the new size of the shape.

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

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

  • Not understanding the given information
  • Not identifying the scale factor correctly
  • Not applying the scale factor correctly
  • Not checking the units of measurement

Q: How do I determine the scale factor in a problem?

A: To determine the scale factor in a problem, you need to look for clues in the problem statement. Some common clues include:

  • A ratio or proportion that compares the size of the original shape to the size of the transformed shape
  • A statement that says "the shape is scaled up by a factor of" or "the shape is scaled down by a factor of"
  • A diagram or picture that shows the original shape and the transformed shape

Q: Can I use dilation and scale factors to solve problems involving 3D shapes?

A: Yes, you can use dilation and scale factors to solve problems involving 3D shapes. However, you need to be careful when working with 3D shapes, as the scale factor may affect the dimensions of the shape in different ways.

Q: How do I check my work when working with dilation and scale factors?

A: To check your work when working with dilation and scale factors, you need to:

  • Read the problem carefully and make sure you understand the given information
  • Check your calculations and make sure you are applying the scale factor correctly
  • Check the units of measurement to make sure they are consistent
  • Check your answer to make sure it makes sense in the context of the problem

Conclusion

Dilation and scale factors are fundamental concepts in mathematics that involve changing the size of a shape or a figure. By understanding dilation and scale factors, you can solve a wide range of problems involving geometry and measurement. In this article, we provided a Q&A guide to help you understand dilation and scale factors better.

Practice Problems

  1. A square is dilated by a scale factor of 2. If the length of one side of the original square is 6 cm, what is the length of one side of the dilated square?
  2. A rectangle is dilated by a scale factor of 3. If the length of one side of the original rectangle is 8 cm, what is the length of one side of the dilated rectangle?
  3. A triangle is dilated by a scale factor of 4. If the length of one side of the original triangle is 12 cm, what is the length of one side of the dilated triangle?

Answer Key

  1. 12 cm
  2. 24 cm
  3. 48 cm

Additional Resources

  • Khan Academy: Dilation and Scale Factors
  • Math Open Reference: Dilation and Scale Factors
  • IXL: Dilation and Scale Factors