What are the two whole numbers closest to the square root of 162?
The Correct Answer and Explanation is:
To determine the two whole numbers closest to the square root of 162, we begin by calculating the square root itself:
[
\sqrt{162} \approx 12.7279
]
Finding the Closest Whole Numbers
The closest whole numbers to 12.7279 are 12 and 13. To confirm this, we can square these numbers to see how they relate to 162:
- (12^2 = 144)
- (13^2 = 169)
Comparison
Now, let’s analyze the results:
- Square of 12:
[
12^2 = 144
]
This is less than 162, which indicates that the square root of 162 is greater than 12. - Square of 13:
[
13^2 = 169
]
This is greater than 162, which indicates that the square root of 162 is less than 13.
Since (12^2 < 162 < 13^2), we confirm that the square root of 162 lies between 12 and 13.
Estimation of Square Root
To gain a better understanding of the proximity of the square root of 162 to these whole numbers, we can further approximate it:
- Between 12 and 13: Since (12.5^2 = 156.25) is also less than 162, we know (\sqrt{162}) is greater than 12.5.
- Closer to 13: Since (12.8^2 = 163.84) is greater than 162, it indicates that (\sqrt{162}) is less than 12.8.
Conclusion
In conclusion, the square root of 162 is approximately 12.7279, placing it clearly between the whole numbers 12 and 13. Thus, the two whole numbers closest to the square root of 162 are 12 and 13. This method illustrates how squaring whole numbers can help determine the range within which a square root lies, aiding in mathematical estimations and calculations.