Problem Analysis:
The task involves estimating square roots of given numbers and determining which two integers each square root lies between. Additionally, you need to determine which integer the square root is closer to. Finally, you are asked to plot these square roots on a number line.
Given Numbers:
1. \( \sqrt{18} \)
2. \( \sqrt{120} \)
3. \( \sqrt{34} \)
Solution:
#### Step 1: Estimate \( \sqrt{18} \)
-
Find the perfect squares around 18:
- \( 4^2 = 16 \)
- \( 5^2 = 25 \)
- Since \( 16 < 18 < 25 \), \( \sqrt{18} \) lies between 4 and 5.
-
Determine which integer it is closer to:
- Calculate the differences:
- \( 18 - 16 = 2 \)
- \( 25 - 18 = 7 \)
- Since 2 is smaller than 7, \( \sqrt{18} \) is closer to 4.
#### Step 2: Estimate \( \sqrt{120} \)
-
Find the perfect squares around 120:
- \( 10^2 = 100 \)
- \( 11^2 = 121 \)
- Since \( 100 < 120 < 121 \), \( \sqrt{120} \) lies between 10 and 11.
-
Determine which integer it is closer to:
- Calculate the differences:
- \( 120 - 100 = 20 \)
- \( 121 - 120 = 1 \)
- Since 1 is smaller than 20, \( \sqrt{120} \) is closer to 11.
#### Step 3: Estimate \( \sqrt{34} \)
-
Find the perfect squares around 34:
- \( 5^2 = 25 \)
- \( 6^2 = 36 \)
- Since \( 25 < 34 < 36 \), \( \sqrt{34} \) lies between 5 and 6.
-
Determine which integer it is closer to:
- Calculate the differences:
- \( 34 - 25 = 9 \)
- \( 36 - 34 = 2 \)
- Since 2 is smaller than 9, \( \sqrt{34} \) is closer to 6.
Final Answers:
1. \( \sqrt{18} \) is between
4 and
5, and it is closer to
4.
2. \( \sqrt{120} \) is between
10 and
11, and it is closer to
11.
3. \( \sqrt{34} \) is between
5 and
6, and it is closer to
6.
Plotting on the Number Line:
-
\( \sqrt{18} \): Between 4 and 5, closer to 4.
-
\( \sqrt{120} \): Between 10 and 11, closer to 11.
-
\( \sqrt{34} \): Between 5 and 6, closer to 6.
Final Answer:
\[
\boxed{
\begin{aligned}
1. & \quad \sqrt{18} \text{ is between } 4 \text{ and } 5, \text{ closer to } 4. \\
2. & \quad \sqrt{120} \text{ is between } 10 \text{ and } 11, \text{ closer to } 11. \\
3. & \quad \sqrt{34} \text{ is between } 5 \text{ and } 6, \text{ closer to } 6.
\end{aligned}
}
\]
Parent Tip: Review the logic above to help your child master the concept of estimating irrational square roots worksheet.
