- Define the area-to-side length relationship: \(s = \sqrt{A}\).
- Calculate the square of the bounds for each provided option to compare with the given area \(A = 200\).
- Determine that \(14.1^2 = 198.81\) and \(14.2^2 = 201.64\), confirming that the side length \(s = \sqrt{200}\) lies between \(14.1\) and \(14.2\).
- Conclude the correct statement: \(\boxed{\text{The length is between 14.1 inches and 14.2 inches.}}\)
Explanation
- Analyze the problem and identify the side length.
The area of a square is related to its side length \(s\) by the formula \(A = s^2\). Given that the area \(A\) is \(200\) square inches, we set up the equation:
\[s^2 = 200\]
To find the side length \(s\), we take the positive square root:
\[s = \sqrt{200}\]
We know that \(\sqrt{200}\) is approximately \(14.142\). We can evaluate the given options by comparing their values to \(s\).
- Evaluate the given options to find the true statement.
We test each option by squaring its boundary values and comparing them to \(200\):
- Option 1: The length is less than \(14\) inches.
Calculating \(14^2 = 196\). Since \(196 < 200\), the side length \(s = \sqrt{200}\) must be greater than \(14\). Thus, this statement is false.
- Option 2: The length is between \(19\) inches and \(20\) inches.
Calculating \(19^2 = 361\) and \(20^2 = 400\). Since \(200\) is not between \(361\) and \(400\), this statement is false.
- Option 3: The length is exactly \(100\) inches.
Calculating \(100^2 = 10,000\), which is clearly not \(200\). Thus, this statement is false.
- Option 4: The length is between \(14.1\) inches and \(14.2\) inches.
Calculating the squares of the bounds:
\(14.1^2 = 198.81\)
\(14.2^2 = 201.64\)
Since \(198.81 < 200 < 201.64\), it follows that \(\sqrt{198.81} < \sqrt{200} < \sqrt{201.64}\), which simplifies to \(14.1 < s < 14.2\). Thus, this statement is true.
- Conclude the correct statement.
Based on our calculations, the length of the side of the square is between \(14.1\) inches and \(14.2\) inches.
The final answer is \(\boxed{\text{The length is between 14.1 inches and 14.2 inches.}}\)
Examples
Suppose you are a carpenter building a custom tabletop that needs to cover exactly 200 square inches of surface area. By calculating the square root of 200, you can determine that each side of the square tabletop must measure approximately 14.14 inches. Knowing this precise range helps you cut your materials accurately to fit your design requirements without wasting wood.