Question
estimate between two ____ numbers
estimate further to the ____ place
3.7^2 < 14 < ____^2
√14 is between _____ and 3.75
estimate further to the ____ place
3.7^2 < 14 < ____^2
√14 is between _____ and 3.75
Answers
GPT-4o mini
To estimate \(\sqrt{14}\), we can start with the given information and fill in the blanks.
1. "estimate between two **whole** numbers" - \(\sqrt{14}\) is between 3 and 4 because \(3^2 = 9\) and \(4^2 = 16\).
2. "estimate further to the **decimal** place" - We can estimate the decimal further. We know it's between 3 and 4.
3. For the inequality \(3.7^2 < 14 < \_\_\_^2\), we calculate \(3.7^2\):
\[
3.7^2 = 13.69
\]
Hence, \(13.69 < 14\) is true. For \(14 < x^2\), we need to find \(x\) such that \(x^2 = 16\), so \(x\) would have to be slightly more than 4 (the nearest perfect square). Thus, we can use:
\[
\sqrt{14} < 4
\]
So, we can fill in this blank with **4**.
4. Finally, "√14 is between **3.74** and 3.75". We can estimate \(\sqrt{14}\) to two decimal places as approximately 3.74 or 3.75. The more precise calculation shows:
\[
3.74^2 = 13.9876\quad \text{and} \quad 3.75^2 = 14.0625
\]
Hence, \(\sqrt{14}\) is indeed between \(3.74\) and \(3.75\).
Putting it all together, we have:
- estimate between two **whole** numbers: 3 and 4
- estimate further to the **decimal** place: 3.74 and 3.75
- \(3.7^2 < 14 < 4^2\)
- \(\sqrt{14}\) is between **3.74** and **3.75**.
1. "estimate between two **whole** numbers" - \(\sqrt{14}\) is between 3 and 4 because \(3^2 = 9\) and \(4^2 = 16\).
2. "estimate further to the **decimal** place" - We can estimate the decimal further. We know it's between 3 and 4.
3. For the inequality \(3.7^2 < 14 < \_\_\_^2\), we calculate \(3.7^2\):
\[
3.7^2 = 13.69
\]
Hence, \(13.69 < 14\) is true. For \(14 < x^2\), we need to find \(x\) such that \(x^2 = 16\), so \(x\) would have to be slightly more than 4 (the nearest perfect square). Thus, we can use:
\[
\sqrt{14} < 4
\]
So, we can fill in this blank with **4**.
4. Finally, "√14 is between **3.74** and 3.75". We can estimate \(\sqrt{14}\) to two decimal places as approximately 3.74 or 3.75. The more precise calculation shows:
\[
3.74^2 = 13.9876\quad \text{and} \quad 3.75^2 = 14.0625
\]
Hence, \(\sqrt{14}\) is indeed between \(3.74\) and \(3.75\).
Putting it all together, we have:
- estimate between two **whole** numbers: 3 and 4
- estimate further to the **decimal** place: 3.74 and 3.75
- \(3.7^2 < 14 < 4^2\)
- \(\sqrt{14}\) is between **3.74** and **3.75**.