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

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

Options:

Correct Answer: 5√3

Solution:

Using tan(30°) = height/shadow, height = 10 * tan(30°) = 10 * (1/√3) = 10√3/3.

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

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 angle of elevation of the sun, which is given as 30°.
Step 3: Recognize that the shadow of the tree is the adjacent side of the triangle, which is 10 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: Substitute the known values into the formula: tan(30°) = height/10.
Step 7: Rearrange the formula to solve for height: height = 10 * tan(30°).
Step 8: Calculate tan(30°), which is equal to 1/√3.
Step 9: Substitute tan(30°) back into the equation: height = 10 * (1/√3).
Step 10: Simplify the expression: height = 10/√3, which can be further simplified to height = 10√3/3.

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.