Using Trigonometry to Find the Lighthouse Height

To solve this problem, we will use trigonometric relationships in a right triangle. In this scenario, the fisherman forms a right triangle with the lighthouse where: - The horizontal distance from the fisherman to the lighthouse is one leg of the triangle. - The height of the lighthouse is the other leg of the triangle. - The angle of elevation from the fisherman to the top of the lighthouse is given. Let's summarize the given information: - Horizontal distance (adjacent side) = 129 meters - Angle of elevation = 36 degrees We need to find the height of the lighthouse (opposite side). We will use the tangent (tan) function in trigonometry, which relates the angle of a right triangle to the ratio of the opposite side to the adjacent side: \[ \tan(\text{angle}) = \frac{\text{opposite}}{\text{adjacent}} \] In this case: \[ \tan(36^\circ) = \frac{\text{height of the lighthouse}}{129 \text{ meters}} \] To find the height of the lighthouse, we rearrange the equation to solve for the opposite side (height): \[ \text{height of the lighthouse} = 129 \text{ meters} \times \tan(36^\circ) \] Now, we need to calculate the tangent of 36 degrees. You can use a scientific calculator for this: \[ \tan(36^\circ) \approx 0.7265 \] Substituting this value back into the equation: \[ \text{height of the lighthouse} = 129 \text{ meters} \times 0.7265 \] \[ \text{height of the lighthouse} \approx 93.71685 \text{ meters} \] Rounding to a reasonable precision: \[ \text{height of the lighthouse} \approx 93.72 \text{ meters} \] So, the height of the lighthouse is approximately 93.72 meters.