Question

For each square below, is the side length rational or irrational? A = 81 cm² A = 89 cm²

Answer 1

A is rational if the side length is rational, as the area of a square is found by multiplying the side length by itself.

A = 81 cm²: The side length is 9 cm, which is rational. Therefore, A is rational.

A = 89 cm²: The side length is approximately 9.93 cm, which is irrational. Therefore, A is irrational.

Explaination:

When finding the side length of a square with a given area, we take the square root of the area to find the side length. If the square root is a whole number or a decimal with no repeating pattern, then the side length is rational. However, if the decimal has a repeating pattern or goes on infinitely without repeating, then the side length is irrational.

In our first example, A = 81 cm². Taking the square root of 81 gives us 9 cm, which is a whole number and therefore rational. In our second example, A = 89 cm². Taking the square root of 89 gives us approximately 9.93 cm, which has a repeating pattern in its decimal form (0.993...). This indicates that the decimal goes on infinitely without repeating and therefore the side length is irrational.

In summary, to determine whether the side length of a square with a given area is rational or irrational, we find the square root of the area and check whether it is a whole number or has a repeating decimal pattern. If it's a whole number, then the side length is rational. If it's not a whole number and has a repeating decimal pattern, then the side length is irrational.

Answer 2

- For the square with area
`\(A = 81 \, \text{cm}^2\)`, the side length is rational (9 cm).

- For the square with area
`\(A = 89 \, \text{cm}^2\)`, the side length is irrational (
`√(89)\)` cm).

To determine whether the side length of a square is rational or irrational, we can use the formula for the area of a square:

`\[ A = \text{side length} * \text{side length} \]`

1. For the square with area
`\(A = 81 \, \text{cm}^2\)`:

Let
`\( s \)` be the side length.

`\( 81 = s * s \)`

`\( s^2 = 81 \)`

`\( s = √(81) \)`

`\( s = 9 \)`

The side length is rational (9 cm).

2. For the square with area
`\(A = 89 \, \text{cm}^2\)`:

Let
`\( s \)` be the side length.

`\( 89 = s * s \)`

`\( s^2 = 89 \)`

`\( s = √(89) \)`

The side length is irrational because
`\(√(89)\)` cannot be simplified to a rational number.