A tree casts a shadow of 10 meters when the angle of elevation of the sun is 30

A tree casts a shadow of 10 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 10 meters when the angle of elevation of the sun is 30 degrees. What is the height of the tree?

Correct Answer: 17.32 meters

Step 1: Understand that the problem involves a right triangle formed by the tree, its shadow, and the line from the top of the tree to the tip of the shadow.
Step 2: Identify the components of the triangle: the height of the tree is the opposite side, the shadow is the adjacent side, and the angle of elevation of the sun is 30 degrees.
Step 3: Recall the trigonometric function tangent (tan), which relates the opposite side to the adjacent side in a right triangle: tan(angle) = opposite/adjacent.
Step 4: Rearrange the formula to find the height of the tree: height = shadow * tan(angle).
Step 5: Substitute the values into the formula: height = 10 meters * tan(30 degrees).
Step 6: Calculate tan(30 degrees), which is equal to √3/3 or approximately 0.577.
Step 7: Multiply the shadow length by the tangent value: height = 10 * (√3/3) = 10 * 0.577 = 5.77 meters.
Step 8: Note that the short solution provided in the question is incorrect; the correct height of the tree is approximately 5.77 meters.

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.