A tree casts a shadow of 20 meters when the angle of elevation of the sun is 30

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Question: A tree casts a shadow of 20 meters when the angle of elevation of the sun is 30 degrees. What is the height of the tree?

Options:

Correct Answer: 20√3 meters

Solution:

Height = shadow * tan(angle) = 20 * tan(30°) = 20 * (1/√3) = 20√3 meters.

A tree casts a shadow of 20 meters when the angle of elevation of the sun is 30 degrees. What is the height of the tree?

A tree casts a shadow of 20 meters when the angle of elevation of the sun is 30 degrees. What is the height of the tree?

Step 1: Understand that the tree, the shadow, and the line from the top of the tree to the tip of the shadow form a right triangle.
Step 2: Identify the angle of elevation of the sun, which is given as 30 degrees.
Step 3: Recognize that the shadow of the tree is the base of the triangle, which is 20 meters long.
Step 4: The height of the tree is the opposite side of the triangle, which we need to find.
Step 5: Use the tangent function, which relates the opposite side (height of the tree) to the adjacent side (length of the shadow) using the formula: tan(angle) = opposite/adjacent.
Step 6: Rearrange the formula to find the height: height = shadow * tan(angle).
Step 7: Substitute the values into the formula: height = 20 * tan(30 degrees).
Step 8: Calculate tan(30 degrees), which is 1/√3.
Step 9: Multiply: height = 20 * (1/√3).
Step 10: Simplify the expression: height = 20/√3, which can also be written as 20√3 meters.

Trigonometry – The problem involves using the tangent function to relate the height of the tree to the length of its shadow based on the angle of elevation of the sun.
Angle of Elevation – Understanding how the angle of elevation affects the relationship between the height of an object and the length of its shadow.