A tree casts a shadow of 10 meters long. If 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 10 meters long. If the angle of elevation of the sun is 30 degrees, what is the height of the tree?

Solution: Height = shadow * tan(angle) = 10 * √3 = 5√3 meters.

Steps: 10

Step 1: Understand that the tree, the ground, and the shadow form a right triangle.
Step 2: Identify the parts of the triangle: the height of the tree is the opposite side, the length of the shadow is the adjacent side, and the angle of elevation is 30 degrees.
Step 3: Recall the tangent function in trigonometry: tan(angle) = opposite/adjacent.
Step 4: For our triangle, this means tan(30 degrees) = height of the tree / length of the shadow.
Step 5: We know the length of the shadow is 10 meters, so we can write: tan(30 degrees) = height / 10.
Step 6: The value of tan(30 degrees) is √3 / 3.
Step 7: Substitute this value into the equation: √3 / 3 = height / 10.
Step 8: To find the height, multiply both sides by 10: height = 10 * (√3 / 3).
Step 9: Simplify the equation: height = 10√3 / 3 meters.
Step 10: To express it in a simpler form, we can approximate or leave it as is.