A sailboat is approaching a cliff. The angle of elevation from the sailboat to the top of the cliff is 35 degrees. The height of the cliff is known to be about 2000 m. How far is the sailboat away from the base of the cliff? Round your answer to 1 decimal place.
Answer (2)
Define the distance between the sailboat and the base of the cliff as d .
Use the tangent function to relate the angle of elevation, the height of the cliff, and the distance d : tan ( 3 5 ∘ ) = d 2000 .
Solve for d : d = t a n ( 3 5 ∘ ) 2000 .
Calculate the value of d and round to 1 decimal place: 2856.3 meters .
Explanation
Problem Setup We are given the angle of elevation from a sailboat to the top of a cliff, which is 35 degrees. The height of the cliff is 2000 meters. We need to find the horizontal distance between the sailboat and the base of the cliff. Let's call this distance d .
Applying Tangent Function We can use the tangent function to relate the angle of elevation, the height of the cliff, and the distance d . The tangent of an angle in a right triangle is the ratio of the opposite side to the adjacent side. In this case, the opposite side is the height of the cliff (2000 m), and the adjacent side is the distance d we want to find. So, we have: tan ( 3 5 ∘ ) = d 2000
Solving for d Now, we need to solve for d . We can rearrange the equation as follows: d = tan ( 3 5 ∘ ) 2000
Calculating d We know that tan ( 3 5 ∘ ) ≈ 0.7002 . Therefore, d = 0.7002 2000 ≈ 2856.296
Final Answer Rounding the distance to 1 decimal place, we get: d ≈ 2856.3 meters
Examples
Understanding angles of elevation is crucial in many real-world scenarios, such as navigation and surveying. For instance, sailors use sextants to measure the angle of elevation of celestial bodies to determine their position at sea. Similarly, surveyors use theodolites to measure angles of elevation and depression to calculate distances and heights in land surveying. These techniques rely on trigonometric principles to accurately map and navigate the world around us.
The distance from the sailboat to the base of the cliff is approximately 2856.3 meters. This is calculated using the tangent function related to the angle of elevation and the height of the cliff. Using the formula d = t a n ( 3 5 ∘ ) 2000 , we find the value for d .
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