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In Example 5, a problem is solved that involves calculating the height of a tower based on the lengths of its shadow when the sun is at different altitudes. Two right triangles are used to set up equations based on the tangent function.
Example 5 deals with a geometric problem surrounding a tower's height and the length of its shadow under two different angles of elevation of the sun (30° and 60°). The solution involves using trigonometric principles, specifically the tangent function, to relate the height of the tower (AB) to the lengths of its shadows (BC and BD) at the specified angles. The relationship established shows that the shadow length at 30° is 40 meters longer than that at 60°, leading to the formulation of two equations based on right triangles formed in the diagram. The mathematical approach culminates in determining that the height of the tower is 20√3 meters.
Height of a Tower: The vertical distance from the base of the tower to its top.
Shadow Length: The distance from the base of the object (the tower) to the tip of its shadow.
Angles of Elevation: The angles formed with the horizontal line to the line of sight to the top of the tower, measured at 30° and 60° in this case.
Trigonometric Functions: Functions such as tangent that relate angles to ratios of sides in triangles.
When the sun is high and shadows low, the tower's height will surely show.
Imagine a brave knight measuring tower heights with a magic shadow stick that grows as the sun bows lower in the sky.
Remember HATS: Height = Adjacent × Tangent of Angle for shadow problems.
Example 1: If a person stands 10 m away from a tree that is 5 m tall, what is the angle of elevation to the top of the tree?
Example 2: A flagpole casts a shadow of length 15 m when the angle of elevation is 45°. What is the height of the flagpole?
Term: Angle of Elevation
Definition: The angle formed between the horizontal line and the line of sight to an object above the horizontal.
The angle formed between the horizontal line and the line of sight to an object above the horizontal.
Term: Tangent Function
Definition: A trigonometric function that relates the angle of a right triangle to the ratio of the length of the opposite side to the length of the adjacent side.
A trigonometric function that relates the angle of a right triangle to the ratio of the length of the opposite side to the length of the adjacent side.
Term: Right Triangle
Definition: A triangle where one angle is exactly 90 degrees.
A triangle where one angle is exactly 90 degrees.
Term: Shadow Length
Definition: The length of the projection of an object, like a tower, cast on the ground due to light, such as sunlight.
The length of the projection of an object, like a tower, cast on the ground due to light, such as sunlight.
Term: Trigonometric Relationship
Definition: Relationships among the angles and sides of triangles.
Relationships among the angles and sides of triangles.