Question: Use trial and improvement to find the approximate value of $\sqrt[3]{278}$. Give your answer to the nearest integer.
1. **State the problem:** We want to find the approximate value of the cube root of 278, written as $\sqrt[3]{278}$, using trial and improvement.
2. **Recall the cube root definition:** The cube root of a number $x$ is a number $y$ such that $y^3 = x$.
3. **Trial and improvement method:** We try integers near the cube root and check their cubes.
4. **Try $6^3$:**
$$6^3 = 6 \times 6 \times 6 = 216$$
Since $216 < 278$, the cube root is greater than 6.
5. **Try $7^3$:**
$$7^3 = 7 \times 7 \times 7 = 343$$
Since $343 > 278$, the cube root is less than 7.
6. **Compare distances:**
$278 - 216 = 62$ and $343 - 278 = 65$.
Since 62 is slightly less than 65, $\sqrt[3]{278}$ is closer to 6 than to 7.
7. **Conclusion:** The nearest integer to $\sqrt[3]{278}$ is $6$.
**Final answer:** $\boxed{6}$