1. **Problem:** Calculate the length of the side marked $x$ in a right triangle with angle $40^\circ$ and adjacent side length $13$ cm. 2. **Formula:** We use the sine function because sine relates the opposite side to the hypotenuse: $$\sin \theta = \frac{\text{Opposite}}{\text{Hypotenuse}}$$ 3. **Important note:** Here, we have the adjacent side and angle, but sine requires opposite and hypotenuse. We need to find the hypotenuse first using cosine: $$\cos 40^\circ = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{13}{H}$$ 4. Solve for hypotenuse $H$: $$H = \frac{13}{\cos 40^\circ}$$ 5. Calculate $H$: $$H = \frac{13}{\cos 40^\circ} = \frac{13}{0.7660} = 16.97 \text{ cm}$$ 6. Now use sine to find $x$: $$\sin 40^\circ = \frac{x}{16.97}$$ 7. Solve for $x$: $$x = 16.97 \times \sin 40^\circ$$ 8. Calculate $x$: $$x = 16.97 \times 0.6428 = 10.91 \text{ cm}$$ 9. **Final answer:** The length of side $x$ is $10.9$ cm (to 1 decimal place).