A pine tree stands 10 meters tall but due to a windstorm it is leaning at a 60 degree angle with the ground. Find an exact expression for the length of the shadow of the tree when the sun is directly overhead.
To find the length of the shadow of the pine tree, we need to use the concept of trigonometry. Let's consider the situation:
The height of the pine tree (opposite side) is 10 meters and the angle between the tree and the ground (angle of elevation) is 60 degrees. We want to find the length of the shadow (adjacent side) when the sun is directly overhead.
Since the sun is directly overhead, we can consider the sun's rays as parallel to the ground. Therefore, we have a right triangle formed by the height of the tree, the length of the shadow, and the ground.
In a right triangle, we can use the tangent function to relate the opposite and adjacent sides:
tan(angle) = opposite/adjacent
Applying this to our situation:
tan(60 degrees) = 10/length of shadow
Now, we can solve for the length of the shadow:
length of shadow = 10/tan(60 degrees)
To find an exact expression for this, we can simplify it further. Remember that tan(60 degrees) is equal to the square root of 3.
length of shadow = 10/(√3)
So, the exact expression for the length of the shadow of the pine tree when the sun is directly overhead is 10/(√3) meters.