1. **Problem statement:** We need to find the length of side $AB$ in the triangular prism. Given are angle $\angle F = 24^\circ$, side $FE = 6$ cm, and slant height $EC = 13$ cm. 2. **Understanding the triangle:** The triangle $ABF$ is right-angled at $F$. We know $FE = 6$ cm is the horizontal leg adjacent to angle $F$, and $AF$ is the vertical leg opposite angle $F$. We want to find $AB$, the hypotenuse of triangle $ABF$. 3. **Using trigonometry:** In right triangle $ABF$, the cosine of angle $F$ relates the adjacent side $FE$ and hypotenuse $AB$: $$\cos(24^\circ) = \frac{FE}{AB}$$ Rearranged to find $AB$: $$AB = \frac{FE}{\cos(24^\circ)}$$ 4. **Calculate $AB$:** $$AB = \frac{6}{\cos(24^\circ)}$$ Using $\cos(24^\circ) \approx 0.9135$: $$AB = \frac{6}{0.9135} \approx 6.57$$ 5. **Final answer:** The length of $AB$ is approximately $6.57$ cm to 2 decimal places.