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. What is the height of 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. What is the height of the tree?

A tree casts a shadow of 15 meters when the angle of elevation of the sun is 30 degrees. 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 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: Rewrite 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.
Step 10: To make it neater, multiply the numerator and denominator by √3: height = (15√3)/3 = 5√3 meters.

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