1. **State the problem:** Ramon has two wood pieces 7 inches and 3 inches long, and he wants to cut a third piece that is the longest side to form an acute triangle. We need to find the length of this longest side so the triangle is acute. 2. **Recall the triangle classification rule based on side lengths:** - Let the sides be $a=3$, $b=7$, and $c$ (the longest side). - The triangle is acute if and only if $$a^2 + b^2 > c^2.$$ - It is right if $$a^2 + b^2 = c^2,$$ - It is obtuse if $$a^2 + b^2 < c^2.$$ 3. **Calculate $a^2 + b^2$:** $$3^2 + 7^2 = 9 + 49 = 58.$$ 4. **Apply the acute condition:** For the triangle to be acute, we need $$58 > c^2 \\ \implies c < \sqrt{58}.$$ 5. **Check the triangle inequality:** The length $c$ must also satisfy the triangle inequality with the other sides: $$c < a + b = 3 + 7 = 10,$$ and $$c > b = 7$$ because $c$ is the longest side. 6. **Combine all constraints:** - Since $c$ is longest, $c > 7$. - For acute triangle, $c < \sqrt{58} \approx 7.62.$ - Also, $c < 10$ from triangle inequality. 7. **Conclusion:** The longest side $c$ must satisfy $$7 < c < \sqrt{58}$$ for the triangle to be acute. **Answer:** The length must be less than $\sqrt{58}$ inches and greater than 7 inches.