1. **Problem statement:** Jane wants to find the length of the hypotenuse $|AC|$ of the right triangle $ABC$ where $AB=2$ m and $BC=3$ m, and then find the angle $\theta$ at $C$. 2. **Using Pythagoras' theorem:** The theorem states that in a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides: $$|AC|^2 = |AB|^2 + |BC|^2$$ 3. **Substitute the known lengths:** $$|AC|^2 = 2^2 + 3^2 = 4 + 9 = 13$$ 4. **Find $|AC|$ in surd form:** $$|AC| = \sqrt{13}$$ 5. **Find angle $\theta$ using trigonometry:** Since $\theta$ is the angle of elevation at $C$, we use the tangent function: $$\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{|AB|}{|BC|} = \frac{2}{3}$$ 6. **Calculate $\theta$:** $$\theta = \tan^{-1}\left(\frac{2}{3}\right)$$ Using a calculator, $$\theta \approx 33.69^\circ$$ **Final answers:** - Length $|AC| = \sqrt{13}$ m - Angle $\theta \approx 33.69^\circ$ (to two decimal places)