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°. We need to find: a) The height up the wall the ladder reaches. b) The length of the ladder. 2. **Relevant formulas and rules:** - In a right triangle, the side opposite an angle is related to the hypotenuse by the sine function: $$\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$$ - The side adjacent to an angle is related to the hypotenuse by the cosine function: $$\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}$$ - The Pythagorean theorem relates the sides: $$\text{hypotenuse}^2 = \text{opposite}^2 + \text{adjacent}^2$$ 3. **Given:** - Adjacent side (distance from wall) = 2 m - Angle with ground = 68° 4. **Find height (opposite side):** Using tangent or sine, but since adjacent and angle are known, use tangent: $$\tan(68^\circ) = \frac{\text{opposite}}{2}$$ Calculate opposite: $$\text{opposite} = 2 \times \tan(68^\circ)$$ Using a calculator: $$\tan(68^\circ) \approx 2.4751$$ So: $$\text{opposite} = 2 \times 2.4751 = 4.9502 \text{ m}$$ 5. **Find length of ladder (hypotenuse):** Using cosine: $$\cos(68^\circ) = \frac{2}{\text{hypotenuse}}$$ Rearranged: $$\text{hypotenuse} = \frac{2}{\cos(68^\circ)}$$ Calculate cosine: $$\cos(68^\circ) \approx 0.3746$$ So: $$\text{hypotenuse} = \frac{2}{0.3746} \approx 5.34 \text{ m}$$ **Final answers:** a) Height up the wall = $4.95$ m b) Length of the ladder = $5.34$ m