1. **Problem statement:** Calculate the length of the ferry route AB across the river given triangle ABC with $\angle CBA = 90^\circ$, $\angle ACB = 42^\circ$, and $BC = 40$ m. 2. **Formula and rules:** In a right triangle, the sides relate to angles via trigonometric functions. Here, $\triangle ABC$ is right-angled at B, so: $$\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}, \quad \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}$$ where $\theta = 42^\circ$ at point C. 3. **Identify sides:** - $BC = 40$ m (adjacent to angle C) - $AB = ?$ (opposite to angle C) - $AC = ?$ (hypotenuse) 4. **Calculate length AB:** Using $\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$, we get: $$\tan(42^\circ) = \frac{AB}{BC} = \frac{AB}{40}$$ Therefore: $$AB = 40 \times \tan(42^\circ)$$ Calculate $\tan(42^\circ)$: $$\tan(42^\circ) \approx 0.9004$$ So: $$AB = 40 \times 0.9004 = 36.016$$ 5. **Final answer:** The length of the ferry route $AB$ is approximately: $$\boxed{36.0 \text{ meters}}$$