A square rug has an area 94 ft. Write the side length as a square root. Then decide if the side length is a rational number.
The Correct Answer and Explanation is:
To determine the side length of the square rug, we need to use the formula for the area of a square, which is:
[
\text{Area} = s^2
]
where ( s ) represents the side length of the square. Given that the area of the square rug is 94 square feet, we can set up the equation:
[
s^2 = 94
]
To solve for ( s ), we take the square root of both sides of the equation:
[
s = \sqrt{94}
]
So, the side length of the rug is ( \sqrt{94} ) feet.
Is ( \sqrt{94} ) a rational number?
A rational number is a number that can be expressed as the quotient of two integers (i.e., as ( \frac{a}{b} ), where ( a ) and ( b ) are integers and ( b \neq 0 )). A number is irrational if it cannot be expressed as a simple fraction and its decimal form goes on forever without repeating.
To check if ( \sqrt{94} ) is rational, we need to determine whether 94 is a perfect square. A perfect square is a number whose square root is an integer. In the case of 94, we see that:
- ( 9^2 = 81 )
- ( 10^2 = 100 )
Since 94 is not a perfect square (it lies between 81 and 100), ( \sqrt{94} ) is not an integer. Additionally, since it cannot be simplified into a fraction of two integers, ( \sqrt{94} ) is an irrational number.
Conclusion:
The side length of the rug is ( \sqrt{94} ) feet, and this is an irrational number because the square root of 94 cannot be expressed as a rational number or a simple fraction.