A lighthouse BC of unknown height stands on the edge of a cliff AB. From a point O, 2500 ft from the base of the cliff, an observer measures the angles of elevation of the top and bottom of the lighthouse and finds them to be 15°40' and 12°13'. How high is the lighthouse.
2500 (tan15°40' - tan12°13') feet
= 160 feet
To solve this problem, we can use trigonometric functions and the concept of similar triangles. Let's break it down step by step:
Step 1: Draw a diagram:
B
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A-------C
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O
Here, AB represents the height of the cliff, BC represents the height of the lighthouse, and OC represents the distance from the observer O to the base of the lighthouse.
Step 2: Identify the given information:
- Angle of elevation from point O to the top of the lighthouse (angle AOC) = 15°40' (15 degrees 40 minutes)
- Angle of elevation from point O to the bottom of the lighthouse (angle BOC) = 12°13' (12 degrees 13 minutes)
- Distance from O to the base of the cliff (OC) = 2500 ft
Step 3: Set up equations using trigonometric functions:
We can use tangent (tan) since we have an angle and a length. The tangent function relates the sides of a right triangle:
For angle AOC:
tan(AOC) = BC / OC
For angle BOC:
tan(BOC) = AB / OC
Step 4: Convert the given angles from degrees and minutes to decimal form:
To use trigonometric functions, we need to convert the angles from degrees and minutes to decimal form.
15°40' = 15 + 40/60 = 15.67 degrees
12°13' = 12 + 13/60 = 12.22 degrees
Step 5: Solve the equations:
Using the given angle values and trigonometric functions, we have two equations with two unknowns (BC and AB). We can solve them simultaneously.
From tang(AOC) = BC / OC:
tan(15.67°) = BC / 2500 ft
From tang(BOC) = AB / OC:
tan(12.22°) = AB / 2500 ft
Step 6: Solve for BC and AB:
Using a scientific calculator, we can find the tangent of each angle and then solve for BC and AB.
BC = tan(15.67°) * 2500 ft
AB = tan(12.22°) * 2500 ft
BC ≈ 647.60 ft
AB ≈ 518.82 ft
Therefore, the height of the lighthouse (BC) is approximately 647.60 feet.