1. **Problem statement:** We want to find the height of a mast on a hill. The hill's height is 542 m above sea level, and the observer is at 365 m above sea level. The angle of elevation to the foot of the mast is $2.87^\circ$ and to the top of the mast is $3.39^\circ$. 2. **Formula and explanation:** We use the tangent function in right triangles: $\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$. Here, the "opposite" side is the vertical height difference from the observer's eye level to the point observed, and the "adjacent" side is the horizontal distance $d$ from the observer to the mast. 3. **Calculate horizontal distance $d$ to the foot of the mast:** Let $h_1 = 542 - 365 = 177$ m be the height difference from observer to foot of mast. Using angle $2.87^\circ$: $$\tan(2.87^\circ) = \frac{177}{d} \implies d = \frac{177}{\tan(2.87^\circ)}$$ 4. **Calculate $d$ numerically:** $$d = \frac{177}{\tan(2.87^\circ)} = \frac{177}{0.0501} \approx 3531.0 \text{ m}$$ 5. **Calculate total height difference $h_2$ to top of mast:** Using angle $3.39^\circ$: $$\tan(3.39^\circ) = \frac{h_2}{d} \implies h_2 = d \times \tan(3.39^\circ)$$ 6. **Calculate $h_2$ numerically:** $$h_2 = 3531.0 \times \tan(3.39^\circ) = 3531.0 \times 0.0592 \approx 209.0 \text{ m}$$ 7. **Calculate mast height:** The mast height $H$ is the difference between $h_2$ and $h_1$: $$H = h_2 - h_1 = 209.0 - 177 = 32.0 \text{ m}$$ **Final answer:** The height of the mast is approximately **32.0 m**.