Question

Compare √42 and √58 plotted on the number line. What is the approximate difference in tenths between the two values?

Answer 1

To compare \(\sqrt{42}\) and \(\sqrt{58}\) and find the difference between their approximate values, we can first estimate each square root.

1. **Estimate \(\sqrt{42}\)**:
- We know that \(6^2 = 36\) and \(7^2 = 49\), so \(6 < \sqrt{42} < 7\).
- To get a closer estimate, we can use approximate values:
\[
\sqrt{42} \approx 6.4\ \text{(since } 6.4^2 = 40.96 \text{, which is close to 42)}
\]

2. **Estimate \(\sqrt{58}\)**:
- We know that \(7^2 = 49\) and \(8^2 = 64\), so \(7 < \sqrt{58} < 8\).
- For a closer approximation:
\[
\sqrt{58} \approx 7.6\ \text{(since } 7.6^2 = 57.76 \text{, which is close to 58)}
\]

Now that we have estimated both square roots:
- \(\sqrt{42} \approx 6.4\)
- \(\sqrt{58} \approx 7.6\)

Next, we can calculate the difference:
\[
\sqrt{58} - \sqrt{42} \approx 7.6 - 6.4 = 1.2
\]

To find the approximate difference in tenths:
- Since \(1.2\) can also be written as \(12\) tenths, the approximate difference in tenths between \(\sqrt{58}\) and \(\sqrt{42}\) is **12 tenths**.

Thus, the approximate difference in tenths between \(\sqrt{42}\) and \(\sqrt{58}\) is \(\boxed{1.2}\).