Math Problem Statement

From the top of a 311-foot lighthouse, a plane is sighted overhead, and a ship is observed directly below the plane. The angle of elevation of the plane is 25°, and the angle of depression of the ship is 36°. Find a) the distance of the ship from the lighthouse and b) the plane's height above the water. Round to the nearest foot.

Solution

This is a trigonometry problem involving angles of elevation and depression. Let’s solve it step by step.

Known Information:

  1. Height of the lighthouse (hh) = 311 feet
  2. Angle of elevation to the plane = 2525^\circ
  3. Angle of depression to the ship = 3636^\circ

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 3636^\circ.
We use the right triangle formed by the lighthouse height, the horizontal distance to the ship (dsd_s), and the angle.

From tangent: tan(36)=OppositeAdjacent=311ds\tan(36^\circ) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{311}{d_s}

Solve for dsd_s: ds=311tan(36)d_s = \frac{311}{\tan(36^\circ)}

Step 2: Plane's height above the water

For the plane, the angle of elevation is 2525^\circ. We first find the slant height (dpd_p) 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: tan(25)=OppositeAdjacent=height of the plane above the lighthouseds\tan(25^\circ) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{\text{height of the plane above the lighthouse}}{d_s}

We calculate dpd_p and add 311311 (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:

  1. How do we find distances using the sine and cosine functions instead of tangent?
  2. What happens if the angles of elevation and depression are reversed?
  3. How can this problem be visualized graphically with right triangles?
  4. What if the plane's angle of elevation changes to 45°? How does the plane's height change?
  5. 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