Rachel is a surveyor and looks through her theodolite which is 1.5 m high, and sites the top of a tree from 16.2 m away. Rachel determines that the tree is 8.1 m tall. What is the angle of depression from the top of the tree to Rachel's theodolite?(Round your answer to one decimal and put only numbers with no units in the box.)
Question
Rachel is a surveyor and looks through her theodolite which is 1.5 m high, and sites the top of a tree from 16.2 m away. Rachel determines that the tree is 8.1 m tall. What is the angle of depression from the top of the tree to Rachel's theodolite?(Round your answer to one decimal and put only numbers with no units in the box.)
Solution
To solve this problem, we can use the tangent of the angle of depression. The tangent of an angle in a right triangle is the ratio of the opposite side to the adjacent side.
Here, the opposite side is the difference in height between the tree and the theodolite (8.1 m - 1.5 m = 6.6 m), and the adjacent side is the distance from Rachel to the tree (16.2 m).
So, the tangent of the angle of depression is 6.6 m / 16.2 m = 0.4074.
To find the angle itself, we use the inverse tangent function (also called the arctangent), which gives the angle whose tangent is a given number.
So, the angle of depression is the arctangent of 0.4074, which is approximately 22.4 degrees when rounded to one decimal place.
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