A tree casts a shadow of 10 meters long. If the angle of elevation of the sun is

A tree casts a shadow of 10 meters long. If the angle of elevation of the sun is 30 degrees, what is the height of the tree?

A tree casts a shadow of 10 meters long. If the angle of elevation of the sun is 30 degrees, what is the height of the tree?

Correct Answer: 5√3 meters

Step 1: Understand that the tree, the ground, and the shadow form a right triangle.
Step 2: Identify the parts of the triangle: the height of the tree is the opposite side, the length of the shadow is the adjacent side, and the angle of elevation is 30 degrees.
Step 3: Recall the tangent function in trigonometry: tan(angle) = opposite/adjacent.
Step 4: For our triangle, this means tan(30 degrees) = height of the tree / length of the shadow.
Step 5: We know the length of the shadow is 10 meters, so we can write: tan(30 degrees) = height / 10.
Step 6: The value of tan(30 degrees) is √3 / 3.
Step 7: Substitute this value into the equation: √3 / 3 = height / 10.
Step 8: To find the height, multiply both sides by 10: height = 10 * (√3 / 3).
Step 9: Simplify the equation: height = 10√3 / 3 meters.
Step 10: To express it in a simpler form, we can approximate or leave it as is.

Trigonometry – The problem involves using the tangent function to relate the height of the tree to the length of its shadow and the angle of elevation of the sun.
Right Triangle Properties – The scenario can be visualized as a right triangle where the height of the tree is the opposite side, the shadow is the adjacent side, and the angle of elevation is the angle between the ground and the line of sight to the sun.