Math Problem Statement

Which irrational number represents the length of each side of the squares? The area of the rectangle is 90. You can also cut it into 2 squares

Solution

Let's break down the problem:

  1. Given: The area of the rectangle is 90, and it can be cut into two squares.
  2. Question: We need to find the length of each side of the squares, represented by an irrational number.

Solution:

Let xx be the length of the side of each square.

  • The area of one square is x2x^2.
  • Since the rectangle is cut into two squares, the total area of the rectangle will be the sum of the areas of the two squares, i.e., 2x22x^2.

Given that the area of the rectangle is 90: 2x2=902x^2 = 90

Dividing both sides by 2: x2=45x^2 = 45

Taking the square root of both sides: x=45x = \sqrt{45}

Simplifying 45\sqrt{45}: x=9×5=35x = \sqrt{9 \times 5} = 3\sqrt{5}

Thus, the length of each side of the squares is 353\sqrt{5}, which is an irrational number.

Final Answer:

The length of each side of the squares is 353\sqrt{5}, which is an irrational number.

Would you like further details or explanations? Here are five related questions to think about:

  1. How do we know that 45\sqrt{45} is irrational?
  2. Can we approximate 353\sqrt{5} to a decimal value?
  3. How would the solution change if the rectangle were cut into more than two squares?
  4. What are other examples of irrational numbers that can represent lengths in geometric problems?
  5. How do you determine if a square root simplifies to a rational number?

Tip: When simplifying square roots, always factor the number into its prime components to check if there are perfect squares you can simplify.

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Math Problem Analysis

Mathematical Concepts

Geometry
Algebra
Irrational Numbers

Formulas

Area of a rectangle
Area of a square
Simplifying square roots

Theorems

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Suitable Grade Level

Grades 9-12