A tree casts a shadow of 15 meters when the angle of elevation of the sun is 30

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Question: A tree casts a shadow of 15 meters when the angle of elevation of the sun is 30 degrees. How tall is the tree?

Options:

Correct Answer: 5√3 meters

Solution:

Using the tangent function, tan(30) = height / 15. Therefore, height = 15 * tan(30) = 15 * (1/√3) = 5√3 meters.

A tree casts a shadow of 15 meters when the angle of elevation of the sun is 30 degrees. How tall is the tree?

A tree casts a shadow of 15 meters when the angle of elevation of the sun is 30 degrees. How tall is 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 sun.
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 (15 meters) is the adjacent side of the triangle, and the height of the tree is the opposite side.
Step 4: Use the tangent function, which relates the opposite side (height of the tree) to the adjacent side (length of the shadow). The formula is: tan(angle) = opposite / adjacent.
Step 5: Substitute the known values into the formula: tan(30 degrees) = height / 15 meters.
Step 6: Calculate tan(30 degrees), which is equal to 1/√3.
Step 7: Set up the equation: 1/√3 = height / 15.
Step 8: Solve for height by multiplying both sides by 15: height = 15 * (1/√3).
Step 9: Simplify the expression: height = 15/√3, which can be rewritten as 15 * (√3/3) = 5√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.
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 top of the tree.