Math Problem Statement
Solution
This is a trigonometry problem involving angles of elevation and depression. Let’s solve it step by step.
Known Information:
- Height of the lighthouse () = 311 feet
- Angle of elevation to the plane =
- Angle of depression to the ship =
We are tasked to find:
a) Distance of the ship from the lighthouse.
b) Plane's height above the water.
Step 1: Distance of the ship from the lighthouse
From the lighthouse, the angle of depression to the ship is .
We use the right triangle formed by the lighthouse height, the horizontal distance to the ship (), and the angle.
From tangent:
Solve for :
Step 2: Plane's height above the water
For the plane, the angle of elevation is . We first find the slant height () from the top of the lighthouse to the plane, and then use it to calculate the plane's total height above the water.
From tangent:
We calculate and add (the height of the lighthouse) to get the total plane height.
Let me compute the numerical values. math
Given data
h_lighthouse = 311 # height of the lighthouse in feet angle_elevation_plane = 25 # angle of elevation to the plane in degrees angle_depression_ship = 36 # angle of depression to the ship in degrees
Calculate the distance of the ship from the lighthouse (d_s)
d_ship = h_lighthouse / math.tan(math.radians(angle_depression_ship))
Calculate the plane's height above the water
First find the additional height of the plane above the lighthouse
height_plane_above_lighthouse = d_ship * math.tan(math.radians(angle_elevation_plane)) total_height_plane = h_lighthouse + height_plane_above_lighthouse
Round results to the nearest foot
d_ship = round(d_ship) total_height_plane = round(total_height_plane)
d_ship, total_height_plane### Results:
a) The distance of the ship from the lighthouse is 428 feet.
b) The plane's height above the water is 511 feet.
Would you like a detailed breakdown of the calculations or further clarification?
Here are 5 related questions to expand your understanding:
- How do we find distances using the sine and cosine functions instead of tangent?
- What happens if the angles of elevation and depression are reversed?
- How can this problem be visualized graphically with right triangles?
- What if the plane's angle of elevation changes to 45°? How does the plane's height change?
- How is this concept applied in navigation or surveying?
Tip: Always ensure your calculator is in degree mode when working with angles in degrees.
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Math Problem Analysis
Mathematical Concepts
Trigonometry
Angles of Elevation and Depression
Right Triangle Geometry
Formulas
tan(angle) = opposite / adjacent
distance = height / tan(angle)
height = base * tan(angle)
Theorems
Basic Trigonometric Ratios
Suitable Grade Level
Grades 9-11
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