What is the nearest integer to the square root of 85?
The Correct Answer and Explanation is :
The correct answer is: 9
To find the nearest integer to the square root of 85, we can start by estimating the square root. We know that:
- The square root of 81 is 9 (since (9^2 = 81)).
- The square root of 100 is 10 (since (10^2 = 100)).
Since 85 is between 81 and 100, we can infer that (\sqrt{85}) is between 9 and 10.
To get a more precise estimate, we can calculate the squares of numbers between 9 and 10:
- (9.1^2 = 82.81)
- (9.2^2 = 84.64)
- (9.3^2 = 86.49)
From these calculations, we see that (9.1^2 < 85 < 9.3^2), which means (\sqrt{85}) is between 9.1 and 9.3. To further narrow it down, we can test 9.2:
- (9.2^2 = 84.64) (still less than 85)
- (9.3^2 = 86.49) (greater than 85)
This reinforces that (\sqrt{85}) is indeed closer to 9.2 than 9.3.
For a numerical approximation, we can use a calculator to find (\sqrt{85} \approx 9.2195).
Now, determining the nearest integer involves rounding 9.2195. Since the decimal is less than 0.5, we round down to 9.
In conclusion, the nearest integer to the square root of 85 is 9. This method of estimation through squaring integers helps in visualizing the values surrounding the square root, allowing us to determine the nearest whole number accurately. By understanding the relationship between squares and square roots, we can apply similar techniques to other numbers as well.