1. **Problem 1: Height of the sand pile** The sand forms a cone with a base diameter of 8 m, so the radius $r$ is half of that: $r=\frac{8}{2}=4$ m. The angle between the ground and the slope is $28^\circ$. We want to find the height $h$ of the pile. 2. **Formula and explanation:** In the right triangle formed by the height, radius, and slope, the tangent of the angle relates height and radius: $$\tan(\theta) = \frac{h}{r}$$ where $\theta=28^\circ$. 3. **Calculate height:** $$h = r \times \tan(28^\circ) = 4 \times \tan(28^\circ)$$ Using a calculator: $$\tan(28^\circ) \approx 0.5317$$ So, $$h = 4 \times 0.5317 = 2.1268$$ Rounded to the nearest tenth: $$h \approx 2.1 \text{ m}$$ --- 4. **Problem 2: Distance from ladder base to house** The ladder reaches 3.8 m up the house and makes an angle of $71^\circ$ with the ground. We want to find the distance $d$ from the base of the ladder to the house. 5. **Formula and explanation:** In the right triangle, the height is the opposite side, and the distance is the adjacent side to the angle. Using tangent: $$\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{3.8}{d}$$ Rearranged: $$d = \frac{3.8}{\tan(71^\circ)}$$ 6. **Calculate distance:** $$\tan(71^\circ) \approx 2.9042$$ So, $$d = \frac{3.8}{2.9042} \approx 1.308$$ Rounded to the nearest tenth: $$d \approx 1.3 \text{ m}$$ --- 7. **Problem 3: Height reached by guy wire on tower** The wire makes an angle of $56^\circ$ with the ground and is 15.4 m from the base of the tower. We want to find the height $h$ the wire reaches on the tower. 8. **Formula and explanation:** In the right triangle, the height is the opposite side, and the distance from the base is the adjacent side. Using tangent: $$\tan(56^\circ) = \frac{h}{15.4}$$ Rearranged: $$h = 15.4 \times \tan(56^\circ)$$ 9. **Calculate height:** $$\tan(56^\circ) \approx 1.4826$$ So, $$h = 15.4 \times 1.4826 = 22.83$$ Rounded to the nearest meter: $$h \approx 23 \text{ m}$$