- Identify the problem as a right-angled triangle where height is the opposite side and distance is the adjacent side.
- Use the tangent function \(\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}\) to relate the height, distance, and angle of elevation.
- Rearrange the equation to solve for the distance: \(d = \frac{45}{\tan(21^{\circ})}\).
- Match the result to the provided options, concluding the correct choice is \(\boxed{\frac{45}{\tan(21^{\circ})}}\).
Explanation
- Analyze the problem setup using geometry.
To find the distance from the observer to the base of the plateau, we can model the situation using a right-angled triangle. The plateau represents the vertical leg of the triangle with a height of \(h = 45\) meters. The observer is at a certain distance \(d\) from the base of the plateau, which represents the horizontal leg of the triangle. The angle of elevation from the observer to the top of the plateau is given as \(\theta = 21^{\circ}\).
- Relate the variables using trigonometry.
In a right-angled triangle, the tangent of an angle is defined as the ratio of the length of the opposite side to the length of the adjacent side. For this specific scenario:
\[\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{h}{d}\]
Substituting the known values into the equation, we have:
\[\tan(21^{\circ}) = \frac{45}{d}\]
- Solve for the distance and select the correct option.
To find the distance \(d\), we rearrange the equation:
\[d \cdot \tan(21^{\circ}) = 45\]
\[d = \frac{45}{\tan(21^{\circ})}\]
Comparing this result to the given options, it is clear that the distance from the observer to the base of the plateau is \(\frac{45}{\tan(21^{\circ})}\) meters. This corresponds to the second option provided in the problem.
Examples
In surveying or engineering, calculating distances to tall structures like buildings, mountains, or plateaus is a common task. By measuring the angle of elevation from your position to the top of the structure, you can determine your exact distance from it using the tangent function. For instance, if you know a tower is \(100\) meters high and you measure an angle of elevation of \(30^{\circ}\), you can easily calculate your distance using \(\frac{100}{\tan(30^{\circ})}\), which simplifies to \(\frac{100}{1/\sqrt{3}} = 100\sqrt{3} \approx 173.2\) meters. This application demonstrates the practical utility of trigonometry in real-world measurements.