Question
Using a number line, what whole number and rational number to the nearest half is 68−−√ between? Show your work. Responses between 8.5 and 9 between 8.5 and 9 - incorrect between 7.5 and 8 between 7.5 and 8 9 and 9.5 between 9 and 9.5 - no response given between 8 and 8.5 between 8 and 8.5 -
Answers
Answer
can you make the math copy and paste able
Answer
I cant coppy the numbers
Answered by
GPT-4o mini
To find the value of \( \sqrt{68} \) and determine between which whole number and rational number (to the nearest half) it lies, let's start by calculating \( \sqrt{68} \).
1. **Estimate \( \sqrt{68} \)**:
- We know that \( 8^2 = 64 \) and \( 9^2 = 81 \).
- Thus, \( \sqrt{68} \) is between \( 8 \) and \( 9 \).
2. **Get a more precise estimate**:
- Let's calculate \( 8.5^2 \):
\[
8.5^2 = 72.25 \quad (\text{which is greater than } 68)
\]
- Next, calculate \( 8.2^2 \) and \( 8.3^2 \):
\[
8.2^2 = 67.24 \quad (\text{which is less than } 68)
\]
\[
8.3^2 = 68.89 \quad (\text{which is greater than } 68)
\]
- Therefore, \( \sqrt{68} \) is between \( 8.2 \) and \( 8.3 \).
3. **Calculating for half-way marks**:
- The mid-point between \( 8 \) and \( 8.5 \) is \( 8.25 \), and the mid-point between \( 8.5 \) and \( 9 \) is \( 8.75 \).
- Since \( \sqrt{68} \approx 8.246 \) (from our earlier estimates), it is less than \( 8.25 \).
4. **Conclusion**:
- Since \( \sqrt{68} \) is between \( 8.2 \) and \( 8.3 \), we conclude that it is between \( 8 \) and \( 8.5 \).
Therefore, \( \sqrt{68} \) is between **8 and 8.5**.
1. **Estimate \( \sqrt{68} \)**:
- We know that \( 8^2 = 64 \) and \( 9^2 = 81 \).
- Thus, \( \sqrt{68} \) is between \( 8 \) and \( 9 \).
2. **Get a more precise estimate**:
- Let's calculate \( 8.5^2 \):
\[
8.5^2 = 72.25 \quad (\text{which is greater than } 68)
\]
- Next, calculate \( 8.2^2 \) and \( 8.3^2 \):
\[
8.2^2 = 67.24 \quad (\text{which is less than } 68)
\]
\[
8.3^2 = 68.89 \quad (\text{which is greater than } 68)
\]
- Therefore, \( \sqrt{68} \) is between \( 8.2 \) and \( 8.3 \).
3. **Calculating for half-way marks**:
- The mid-point between \( 8 \) and \( 8.5 \) is \( 8.25 \), and the mid-point between \( 8.5 \) and \( 9 \) is \( 8.75 \).
- Since \( \sqrt{68} \approx 8.246 \) (from our earlier estimates), it is less than \( 8.25 \).
4. **Conclusion**:
- Since \( \sqrt{68} \) is between \( 8.2 \) and \( 8.3 \), we conclude that it is between \( 8 \) and \( 8.5 \).
Therefore, \( \sqrt{68} \) is between **8 and 8.5**.
Answered by
GPT-4o mini
Sure! Here’s the math in a copy-and-paste-friendly format:
1. **Estimate**:
- \( 8^2 = 64 \)
- \( 9^2 = 81 \)
- Thus, \( \sqrt{68} \) is between \( 8 \) and \( 9 \).
2. **Calculate \( 8.5^2 \)**:
\[
8.5^2 = 72.25 \quad (\text{greater than } 68)
\]
3. **Calculate \( 8.2^2 \) and \( 8.3^2 \)**:
\[
8.2^2 = 67.24 \quad (\text{less than } 68)
\]
\[
8.3^2 = 68.89 \quad (\text{greater than } 68)
\]
4. **Determine the range**:
- \( \sqrt{68} \) is between \( 8.2 \) and \( 8.3 \).
5. **Halfway points**:
- Mid-point between \( 8 \) and \( 8.5 \) is \( 8.25 \).
- Mid-point between \( 8.5 \) and \( 9 \) is \( 8.75 \).
- Since \( \sqrt{68} \approx 8.246 \) (which is less than \( 8.25 \)):
**Conclusion**:
- \( \sqrt{68} \) is between \( 8 \) and \( 8.5 \).
1. **Estimate**:
- \( 8^2 = 64 \)
- \( 9^2 = 81 \)
- Thus, \( \sqrt{68} \) is between \( 8 \) and \( 9 \).
2. **Calculate \( 8.5^2 \)**:
\[
8.5^2 = 72.25 \quad (\text{greater than } 68)
\]
3. **Calculate \( 8.2^2 \) and \( 8.3^2 \)**:
\[
8.2^2 = 67.24 \quad (\text{less than } 68)
\]
\[
8.3^2 = 68.89 \quad (\text{greater than } 68)
\]
4. **Determine the range**:
- \( \sqrt{68} \) is between \( 8.2 \) and \( 8.3 \).
5. **Halfway points**:
- Mid-point between \( 8 \) and \( 8.5 \) is \( 8.25 \).
- Mid-point between \( 8.5 \) and \( 9 \) is \( 8.75 \).
- Since \( \sqrt{68} \approx 8.246 \) (which is less than \( 8.25 \)):
**Conclusion**:
- \( \sqrt{68} \) is between \( 8 \) and \( 8.5 \).
Answered by
GPT-4o mini
I apologize for the confusion. Here is the mathematics presented in plain text format so you can easily copy and paste the numbers:
1. Estimate:
- 8^2 = 64
- 9^2 = 81
- Thus, √68 is between 8 and 9.
2. Calculate:
- 8.5^2 = 72.25 (greater than 68)
3. Calculate:
- 8.2^2 = 67.24 (less than 68)
- 8.3^2 = 68.89 (greater than 68)
4. Determine the range:
- √68 is between 8.2 and 8.3.
5. Halfway points:
- Mid-point between 8 and 8.5 is 8.25.
- Mid-point between 8.5 and 9 is 8.75.
- Since √68 ≈ 8.246 (which is less than 8.25):
Conclusion:
- √68 is between 8 and 8.5.
Feel free to copy and paste this text!
1. Estimate:
- 8^2 = 64
- 9^2 = 81
- Thus, √68 is between 8 and 9.
2. Calculate:
- 8.5^2 = 72.25 (greater than 68)
3. Calculate:
- 8.2^2 = 67.24 (less than 68)
- 8.3^2 = 68.89 (greater than 68)
4. Determine the range:
- √68 is between 8.2 and 8.3.
5. Halfway points:
- Mid-point between 8 and 8.5 is 8.25.
- Mid-point between 8.5 and 9 is 8.75.
- Since √68 ≈ 8.246 (which is less than 8.25):
Conclusion:
- √68 is between 8 and 8.5.
Feel free to copy and paste this text!
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