From the base of a hill , the angle of elevation is 23 degrees . The tree forms a 110 degrees angle with the hillside . The distance from the base of the hill to the base of the tree is 120 meters. Find the height of the tree.
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To find the height of the tree, you can use the trigonometric function tangent.
Let's draw a right triangle to represent the situation. The base of the triangle will be the distance from the base of the hill to the base of the tree, which is 120 meters. The height of the triangle will represent the height of the tree that we're trying to find.
The angle of elevation from the base of the hill to the top of the tree is 23 degrees, so we can label this angle as A. The angle formed between the hillside and the line connecting the base of the hill to the base of the tree is 110 degrees, so we label this angle as B.
Now, let's apply the tangent function to find the height of the tree.
Tangent of angle A = height of the tree / 120 meters (base of the triangle)
Therefore, we have the equation:
tan(23 degrees) = height / 120 meters
To solve for the height of the tree, we can rearrange the equation:
height = 120 meters * tan(23 degrees)
Now, we can calculate the height using a calculator:
height ≈ 120 meters * 0.4245
height ≈ 50.94 meters
Therefore, the height of the tree is approximately 50.94 meters.