A tree broken over by the wind forms a right triangle with the ground.
If the broken part makes an angle of 50° with the ground and the top
of the tree is now 20 ft from its base, how tall was the tree? Provide
an Illustration.
To solve this problem, we can use trigonometry, specifically the tangent ratio. The tangent of an angle is defined as the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle.
Let's denote the height of the tree as "h" and the distance from the base of the tree to the point where it broke as "x". In this case, we can consider the height of the tree as the opposite side of the angle of 50°, and the distance "x" as the adjacent side. The given distance from the top of the tree to its base, which is 20 ft, will be our hypotenuse.
From the definition of tangent, we have:
tan(50°) = h / x
Now, we can solve for "h" by rearranging the equation:
h = x * tan(50°)
Since we know that the top of the tree is 20 ft from its base, we can substitute this value into the equation:
h = 20 ft * tan(50°)
Using a scientific calculator, find the tangent of 50°, and multiply it by 20 ft to get the height "h" of the tree.
Illustration:
```
|
|\
h | \
| \
| \
| \ 20 ft
| \
| \
|_______\
x
```
In this illustration, the left side of the triangle represents the height of the tree (h), the base is the distance from the top to the point where the tree broke (x), and the hypotenuse is the distance from the top to the base (20 ft).