1. Trigonometric Functions (Engineering): Problem: A construction crane is extending a beam to reach the top of a building. The angle of elevation from the ground to the top of the beam is 40 degrees. If the beam's length is 50 meters, how high is the top of the beam from the ground?
Directions: Be able to draw/illustrate the given problems and show pertinent solutions.
1. Trigonometric Functions (Engineering):
Problem: A construction crane is extending a beam to reach the top of a building. The angle of elevation from the ground to the top of the beam is 40 degrees. If the beam's length is 50 meters, how high is the top of the beam from the ground?
2. Angle of Elevation and Depression (Astronomy):
Problem: A stargazer is observing a star in the night sky. If the angle of elevation from the observer's position to the star is 60 degrees and the observer is 1000 meters above sea level, how far is the star from the observer?
3. Law of Sines and Cosines (Surveying):
Problem: A surveyor is measuring two sides and the included angle of a triangular piece of land. The lengths of the sides are 200 meters and 300 meters, and the included angle is 120 degrees. What is the length of the third side?
4. Ellipse (Navigation):
Problem: A ship is navigating along a route defined by an elliptical path. The ship's current position is known, along with the major and minor axes of the ellipse. Determine the closest point on the ellipse to the ship's current position.
5. Hyperbola (Cartography):
Problem: A cartographer is creating a map projection using a hyperbolic shape. The parameters of the hyperbola, such as the foci and the eccentricity, are given. Determine the coordinates of a specific point on the hyperbola, which will be used as a reference point for map construction.
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