1. **State the problem:** We need to find the length of side $c$ in a right triangle where angle $B$ is $35^\circ$, side $BC$ (opposite angle $A$) is $16$ m, and angle $C$ is the right angle. 2. **Identify the sides:** In the triangle, side $BC = 16$ m is opposite angle $A$, side $c$ is the hypotenuse opposite the right angle $C$, and angle $B = 35^\circ$. 3. **Use trigonometric ratios:** Since $c$ is the hypotenuse and $BC$ is the side opposite angle $A$, but we know angle $B$, we can use the sine function relative to angle $B$: $$\sin(B) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{BC}{c}$$ 4. **Plug in known values:** $$\sin(35^\circ) = \frac{16}{c}$$ 5. **Solve for $c$:** $$c = \frac{16}{\sin(35^\circ)}$$ 6. **Calculate the value:** $$c = \frac{16}{0.574} \approx 27.9$$ 7. **Round to the nearest integer:** $$c \approx 28$$ **Final answer:** The length of side $c$ is approximately $28$ meters.