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 -

Answer 1

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.

Answer 2

can you make the math copy and paste able

Answer 3

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 \( 8.5 \).

Answer 4

I cant coppy the numbers

Answer 5

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!