1. **State the problem:** We need to find the length of side $x$ in a right triangle where one leg is 6 units and the angle opposite to $x$ is $70^\circ$. 2. **Identify the sides and angle:** The side of length 6 is adjacent to the $70^\circ$ angle, and $x$ is the hypotenuse. 3. **Formula used:** In a right triangle, the cosine of an angle is the ratio of the adjacent side to the hypotenuse: $$\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}$$ 4. **Apply the formula:** $$\cos(70^\circ) = \frac{6}{x}$$ 5. **Solve for $x$:** $$x = \frac{6}{\cos(70^\circ)}$$ 6. **Calculate the cosine:** $$\cos(70^\circ) \approx 0.3420$$ 7. **Substitute and compute:** $$x = \frac{6}{0.3420} \approx 17.54$$ 8. **Final answer:** The length of $x$ is approximately **17.54** units.