Final answer:
To determine the height of a cliff with two different angles of elevation, use trigonometry to solve for the height in a system of two equations derived from tangent ratios. Solve these equations simultaneously to find the height of the cliff.
Step-by-step explanation:
Calculating the Height of a Cliff Using Trigonometry
To determine the height of a cliff using the angles of elevation from two different positions, we can employ trigonometric functions such as the tangent. From the hiker's initial position, the angle of elevation is 24.6 degrees, and after walking 171 feet closer, the angle of elevation becomes 57.6 degrees. To solve this, we will apply the tangent function in two right-angled triangles formed by the hiker's positions and the top of the cliff.
Let's denote the height of the cliff as 'H' and the original distance from the hiker to the base of the cliff as 'D'. After the hiker moves 171 feet closer, the new distance from the hiker to the base of the cliff is 'D - 171'. Using the tangent function:
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- tan(24.6 degrees) = H / D
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- tan(57.6 degrees) = H / (D - 171)
We have two equations with two unknowns (H and D), which can be solved simultaneously to find the cliff height. Once we determine 'D', we can plug it back into either equation to calculate 'H'.
This example showcases the practical application of trigonometry in real-world situations, such as determining the height of geographical features without the need for direct measurement.