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)
Recognize the problem as a right triangle scenario where the cliff height is the opposite side, the distance to the sailboat is the adjacent side, and the angle of elevation is given.
Apply the tangent function: tan ( 3 5 ∘ ) = d 2000 .
Solve for the distance d : d = t a n ( 3 5 ∘ ) 2000 .
Calculate the distance and round to one decimal place: 2856.3 m .
Explanation
Problem analysis We are given a right triangle formed by the cliff (height), the distance from the sailboat to the base of the cliff, and the line of sight from the sailboat to the top of the cliff. The angle of elevation is the angle between the horizontal line (distance from the sailboat to the base) and the line of sight. We are given the height of the cliff as 2000 m and the angle of elevation as 35 degrees. We need to find the distance between the sailboat and the base of the cliff.
Set up the equation Let d be the distance between the sailboat and the base of the cliff. 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 . Therefore, we have: tan ( 3 5 ∘ ) = d 2000
Solve for d Now we solve for d :
d = tan ( 3 5 ∘ ) 2000
Calculate the value of d We know that tan ( 3 5 ∘ ) ≈ 0.7002 . Therefore, d = 0.7002 2000 ≈ 2856.3
Round the answer Rounding to 1 decimal place, we get d ≈ 2856.3 m.
Examples
Imagine you're navigating a ship towards a lighthouse on a coastline. Knowing the height of the lighthouse and measuring the angle of elevation to its top, you can calculate your distance from the shore. This is crucial for safe navigation, helping you avoid running aground or colliding with the coastline. This simple trigonometric calculation ensures a safe journey by providing essential distance information.
The sailboat is approximately 2856.3 m away from the base of the cliff. This distance is calculated using the tangent function relating the height of the cliff and the angle of elevation. The formula used is d = t a n ( 3 5 ∘ ) 2000 .
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