Final answer:
The distance between the tree and the 30 m high tower, with an angle of depression of 30 degrees, is calculated using the tangent function. Upon solving, the distance is found to be approximately 51.99 m, which when rounded, is 52 meters.
Step-by-step explanation:
The student is asking to find the distance between the base of a tree and the top of a 30 m high tower when the angle of depression is 30 degrees. This is a classic trigonometry problem involving right triangles. To find the distance between the tree and the tower, we can use the tangent trigonometric function, tangent of an angle equals opposite side over adjacent side (tan(\(θ\)) = opposite/adjacent).
Since the angle of depression from the top of the tower to the base of the tree is 30 degrees, the same angle applies between the observer's line of sight and the horizontal line. We can consider the height of the tower as the opposite side and the distance between the tree and the tower as the adjacent side. Using the known height (30 m) as the opposite side, we have:
tan(30°) = opposite (30 m) / adjacent (x)
To solve for x, the adjacent side, we remove it from the denominator:
x = 30 m / tan(30°)
Using the tan(30°) value of √3/3 or approximately 0.577, the calculation will be:
x = 30 m / 0.577
This simplifies to approximately x = 51.99 m, which when rounded gives us a distance of 52 meters.
Therefore, the distance between the tree and the tower, rounded to the nearest whole number, is 52 meters.