Final answer:
To calculate the height of a tree with a 60-meter long shadow when the Sun is 30 degrees above the horizon, use the tangent trigonometric function. The tree's height is calculated to be around 34.62 meters tall.
Step-by-step explanation:
The student is asking how to calculate the height of a tree given the length of its shadow when the Sun is 30 degrees above the horizon. This is a trigonometry problem involving a right-angled triangle, where the tree represents the opposite side, the shadow represents the adjacent side, and the angle given is the angle of elevation of the Sun. Since we have the angle and the length of the adjacent side, we can use the tangent trigonometric ratio to find the height of the tree. The tangent of an angle in a right-angled triangle is the ratio of the opposite side to the adjacent side (tangent = opposite/adjacent).
Step-by-Step Solution:
- Write down the tangent function for the angle of elevation: tangent(30 degrees) = height of tree / length of shadow.
- Using the known shadow length (60m) and the tangent of 30 degrees (which is 1/√3 or about 0.577), set up the equation to solve for the tree's height.
- Multiply both sides by 60m to isolate the tree's height: height of tree = 60m * tangent(30 degrees).
- Calculate the height using the value of the tangent of 30 degrees: height of tree = 60m * 0.577.
- Perform the multiplication to find the height: height of tree = 34.62m (rounded to two decimal places).
Therefore, the tree is approximately 34.62 meters tall.