- Find the closest perfect squares to 83, which are \(81\) and \(100\).
- Recognize that \(\sqrt{81} = 9\) and \(\sqrt{100} = 10\).
- Since \(81 < 83 < 100\), the square root \(\sqrt{83}\) is bounded by these integers.
- Conclude that \(\sqrt{83}\) lies between \(\boxed{9 \text{ and } 10}\).
Explanation
- Problem Analysis and Methodology
To determine between which two consecutive whole numbers \(\sqrt{83}\) lies, we need to find two consecutive integers, \(n\) and \(n+1\), such that the square of \(n\) is less than \(83\) and the square of \(n+1\) is greater than \(83\). This is expressed by the inequality \(n^2 < 83 < (n+1)^2\). By identifying the perfect squares nearest to \(83\), we can effectively bound the value of its square root.
- Evaluating Squares and Determining Bounds
We examine perfect squares of integers to find those that bracket \(83\).
- Calculating squares of integers near \(\sqrt{83} \approx 9\):
- \(9^2 = 81\)
- \(10^2 = 100\)
Since \(81 < 83 < 100\), we have confirmed that \(9^2 < 83 < 10^2\). Taking the square root of all parts of this inequality yields \(9 < \sqrt{83} < 10\). Therefore, \(\sqrt{83}\) must lie between the consecutive whole numbers \(9\) and \(10\).
- Conclusion
Since \(\sqrt{81} = 9\) and \(\sqrt{100} = 10\), it is known that \(\sqrt{83}\) is between \(9\) and \(10\). The final answer is \(\boxed{9 \text{ and } 10}\).
Examples
Understanding how to estimate square roots of non-perfect squares is a fundamental skill in geometry and measurement. For example, if you are building a square garden bed and you have a total area of 83 square feet, you can quickly estimate that each side of the garden must be between 9 and 10 feet long, specifically \(\sqrt{83} \approx 9.11\) feet. This helps in quickly calculating the amount of fencing needed without needing a precise calculator immediately.