A tree casts a shadow of 20 m when the angle of elevation of the sun is 30 degrees. What is the height of the tree?

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Q: A tree casts a shadow of 20 m when the angle of elevation of the sun is 30 degrees. What is the height of the tree?

Solution: Using tan(30°) = height/20, we have 1/√3 = height/20. Therefore, height = 20/√3 ≈ 11.55 m.

Steps: 10

Step 1: Understand that the angle of elevation is the angle between the ground and the line from the top of the tree to the sun.
Step 2: Recognize that the shadow of the tree and the height of the tree form a right triangle with the ground.
Step 3: Identify the angle of elevation of the sun, which is given as 30 degrees.
Step 4: Use the tangent function, which relates the angle to the opposite side (height of the tree) and the adjacent side (length of the shadow).
Step 5: Write the equation: tan(30°) = height / shadow length. Here, shadow length is 20 m.
Step 6: Substitute the values into the equation: tan(30°) = height / 20.
Step 7: Recall that tan(30°) is equal to 1/√3.
Step 8: Set up the equation: 1/√3 = height / 20.
Step 9: Solve for height by multiplying both sides by 20: height = 20 / √3.
Step 10: Calculate the height: height ≈ 20 / 1.732 ≈ 11.55 m.