1. **Problem statement:** We have a ladder leaning against a wall forming a right triangle. The foot of the ladder is 2 m from the wall, and the angle between the ladder and the ground is 68°. 2. **What is asked?** (a) Find the height the ladder reaches on the wall. (b) Find the length of the ladder. 3. **Relevant formulas and rules:** - Use trigonometric ratios in a right triangle. - The angle given is between the ladder (hypotenuse) and the ground (adjacent side). - Height corresponds to the side opposite the angle. - Ladder length is the hypotenuse. 4. **Step (a): Find the height (opposite side):** Use the tangent function: $$\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$$ Here, $\theta = 68^\circ$, adjacent = 2 m. Calculate height: $$\text{height} = 2 \times \tan(68^\circ)$$ Using a calculator: $$\tan(68^\circ) \approx 2.4751$$ So, $$\text{height} = 2 \times 2.4751 = 4.9502 \text{ m}$$ 5. **Step (b): Find the ladder length (hypotenuse):** Use the cosine function: $$\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}$$ Rearranged: $$\text{hypotenuse} = \frac{\text{adjacent}}{\cos(\theta)} = \frac{2}{\cos(68^\circ)}$$ Calculate: $$\cos(68^\circ) \approx 0.3746$$ So, $$\text{hypotenuse} = \frac{2}{0.3746} = 5.34 \text{ m}$$ **Final answers:** (a) The ladder reaches approximately $4.95$ m up the wall. (b) The ladder length is approximately $5.34$ m.