Question
if the square root of x has a decimal approximation of 3.4, thenx is between which two integers? In your final answer, include all calculations.
Answers
Answer
can you make it less advanced and without all the symbols and slashes
Answer
can you do it without all the slashes
Answered by
GPT-4o mini
To determine the integer range for \( x \) given that the square root of \( x \) has a decimal approximation of 3.4, we start by expressing this mathematically:
\[
\sqrt{x} \approx 3.4
\]
To find the possible values for \( x \), we will square the approximate value:
\[
x \approx (3.4)^2
\]
Calculating \( (3.4)^2 \):
\[
3.4 \times 3.4 = 11.56
\]
This means that \( x \) is approximately 11.56. Now we need to find the range of integers for \( x \) by considering the range for the square root:
\[
\sqrt{x} \text{ is between } 3.4 - 0.5 \text{ and } 3.4 + 0.5
\]
Calculating these values:
\[
3.4 - 0.5 = 2.9
\]
\[
3.4 + 0.5 = 3.9
\]
Thus, we have:
\[
2.9 < \sqrt{x} < 3.9
\]
Now we will square the entire inequality:
\[
(2.9)^2 < x < (3.9)^2
\]
Calculating \( (2.9)^2 \):
\[
2.9 \times 2.9 = 8.41
\]
Calculating \( (3.9)^2 \):
\[
3.9 \times 3.9 = 15.21
\]
So, we have:
\[
8.41 < x < 15.21
\]
Now, since \( x \) must be an integer, we can conclude that \( x \) lies between the greatest integer less than 8.41 and the smallest integer greater than 15.21. Therefore, the range of integers is:
\[
9 \leq x \leq 15
\]
Thus, \( x \) is between the integers **9 and 15**.
In conclusion, the answer is:
**x is between 9 and 15.**
\[
\sqrt{x} \approx 3.4
\]
To find the possible values for \( x \), we will square the approximate value:
\[
x \approx (3.4)^2
\]
Calculating \( (3.4)^2 \):
\[
3.4 \times 3.4 = 11.56
\]
This means that \( x \) is approximately 11.56. Now we need to find the range of integers for \( x \) by considering the range for the square root:
\[
\sqrt{x} \text{ is between } 3.4 - 0.5 \text{ and } 3.4 + 0.5
\]
Calculating these values:
\[
3.4 - 0.5 = 2.9
\]
\[
3.4 + 0.5 = 3.9
\]
Thus, we have:
\[
2.9 < \sqrt{x} < 3.9
\]
Now we will square the entire inequality:
\[
(2.9)^2 < x < (3.9)^2
\]
Calculating \( (2.9)^2 \):
\[
2.9 \times 2.9 = 8.41
\]
Calculating \( (3.9)^2 \):
\[
3.9 \times 3.9 = 15.21
\]
So, we have:
\[
8.41 < x < 15.21
\]
Now, since \( x \) must be an integer, we can conclude that \( x \) lies between the greatest integer less than 8.41 and the smallest integer greater than 15.21. Therefore, the range of integers is:
\[
9 \leq x \leq 15
\]
Thus, \( x \) is between the integers **9 and 15**.
In conclusion, the answer is:
**x is between 9 and 15.**
Answered by
GPT-4o mini
Sure! Let’s break it down in simpler terms without using symbols.
We know that the square root of a number \( x \) is about 3.4. To find out the range of whole numbers that \( x \) could be, we can follow these steps:
1. **Estimate \( x \):** If the square root of \( x \) is about 3.4, we can find \( x \) by squaring 3.4. Squaring means multiplying the number by itself.
So, we calculate:
\[
3.4 times 3.4 equals 11.56
\]
This means \( x \) is about 11.56.
2. **Find the range:** Since we're working with approximations, let's think about what numbers give square roots close to 3.4. We can say that:
The square root of \( x \) is between 2.9 (that’s 3.4 minus 0.5) and 3.9 (that’s 3.4 plus 0.5).
3. **Square the range:** Now we'll square these two new numbers to find the range for \( x \).
- Squaring 2.9 gives:
\[
2.9 times 2.9 equals 8.41
\]
- Squaring 3.9 gives:
\[
3.9 times 3.9 equals 15.21
\]
This tells us that \( x \) should be more than 8.41 and less than 15.21.
4. **Identify whole numbers:** Since we're looking for whole numbers, we notice that the smallest whole number greater than 8.41 is 9, and the largest whole number less than 15.21 is 15.
So, in conclusion, \( x \) is a whole number that is between **9 and 15**.
We know that the square root of a number \( x \) is about 3.4. To find out the range of whole numbers that \( x \) could be, we can follow these steps:
1. **Estimate \( x \):** If the square root of \( x \) is about 3.4, we can find \( x \) by squaring 3.4. Squaring means multiplying the number by itself.
So, we calculate:
\[
3.4 times 3.4 equals 11.56
\]
This means \( x \) is about 11.56.
2. **Find the range:** Since we're working with approximations, let's think about what numbers give square roots close to 3.4. We can say that:
The square root of \( x \) is between 2.9 (that’s 3.4 minus 0.5) and 3.9 (that’s 3.4 plus 0.5).
3. **Square the range:** Now we'll square these two new numbers to find the range for \( x \).
- Squaring 2.9 gives:
\[
2.9 times 2.9 equals 8.41
\]
- Squaring 3.9 gives:
\[
3.9 times 3.9 equals 15.21
\]
This tells us that \( x \) should be more than 8.41 and less than 15.21.
4. **Identify whole numbers:** Since we're looking for whole numbers, we notice that the smallest whole number greater than 8.41 is 9, and the largest whole number less than 15.21 is 15.
So, in conclusion, \( x \) is a whole number that is between **9 and 15**.
Answered by
GPT-4o mini
Of course! Let's explain everything in very simple terms without any symbols.
1. **Understanding the square root:** We know that the square root of a number \( x \) is approximately 3.4. This means that if you take the square root of \( x \), you get a number that is close to 3.4.
2. **Finding \( x \):** To find out what \( x \) could be, we can take 3.4 and multiply it by itself (this is called squaring). So, when we multiply 3.4 by 3.4, we get about 11.56. This tells us that \( x \) is around 11.56.
3. **Establishing a range:** Since we know square roots can be approximated, let's think of a small range around 3.4. We can say that the square root of \( x \) could be between 2.9 (3.4 minus 0.5) and 3.9 (3.4 plus 0.5).
4. **Squaring the range:** Now, we square both of these new numbers to find the possible values for \( x \).
- When we multiply 2.9 by itself, we get about 8.41.
- When we multiply 3.9 by itself, we get about 15.21.
5. **Finding whole numbers:** Now we know that \( x \) should be greater than 8.41 and less than 15.21. The smallest whole number that is more than 8.41 is 9, and the largest whole number that is less than 15.21 is 15.
Putting it all together, \( x \) is a whole number between 9 and 15.
1. **Understanding the square root:** We know that the square root of a number \( x \) is approximately 3.4. This means that if you take the square root of \( x \), you get a number that is close to 3.4.
2. **Finding \( x \):** To find out what \( x \) could be, we can take 3.4 and multiply it by itself (this is called squaring). So, when we multiply 3.4 by 3.4, we get about 11.56. This tells us that \( x \) is around 11.56.
3. **Establishing a range:** Since we know square roots can be approximated, let's think of a small range around 3.4. We can say that the square root of \( x \) could be between 2.9 (3.4 minus 0.5) and 3.9 (3.4 plus 0.5).
4. **Squaring the range:** Now, we square both of these new numbers to find the possible values for \( x \).
- When we multiply 2.9 by itself, we get about 8.41.
- When we multiply 3.9 by itself, we get about 15.21.
5. **Finding whole numbers:** Now we know that \( x \) should be greater than 8.41 and less than 15.21. The smallest whole number that is more than 8.41 is 9, and the largest whole number that is less than 15.21 is 15.
Putting it all together, \( x \) is a whole number between 9 and 15.
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