To find a rational number with a denominator of 7 that lies between \(7\sqrt{2}\) and \(8\sqrt{2}\), we first need to calculate \(7\sqrt{2}\) and \(8\sqrt{2}\).
-
Calculate \(7\sqrt{2}\): \[ 7\sqrt{2} \approx 7 \times 1.414 \approx 9.898 \]
-
Calculate \(8\sqrt{2}\): \[ 8\sqrt{2} \approx 8 \times 1.414 \approx 11.312 \]
So, we are looking for a rational number \( \frac{n}{7} \) such that:
\[ 9.898 < \frac{n}{7} < 11.312 \]
Multiplying all parts of the inequality by 7 gives:
\[ 9.898 \times 7 < n < 11.312 \times 7 \]
Calculating these:
- \(9.898 \times 7 \approx 69.286\)
- \(11.312 \times 7 \approx 79.184\)
Thus, we need \(n\) such that:
\[ 69.286 < n < 79.184 \]
The integers that satisfy this inequality are \(70, 71, 72, 73, 74, 75, 76, 77, 78, 79\).
Now, we will choose one of these integers to form our rational number. An easy choice is \(n = 70\).
So, the rational number is:
\[ \frac{70}{7} = 10 \]
Verifying:
- \(7\sqrt{2} \approx 9.898 < 10 < 11.312 \approx 8\sqrt{2}\)
This confirms that \( \frac{70}{7} \) is indeed between \(7\sqrt{2}\) and \(8\sqrt{2}\).
Thus, one rational number with a denominator of 7 that is between \(7\sqrt{2}\) and \(8\sqrt{2}\) is:
\[ \frac{70}{7} \]
It can also be expressed as a simplified improper fraction:
\[ \frac{70}{7} = 10 \]