A pine tree casts a 12-meter shadow on level ground. The angle of elevation from the tip of the shadow to the top of the tree is 50 degrees. To the nearest tenth of a meter, what is the height of the pine tree? Explain
To find the height of the pine tree, you can use trigonometry.
First, let's define the angle of elevation and the shadow length. The angle of elevation is the angle between the ground and the line of sight from the tip of the shadow to the top of the tree. The shadow length is the distance between the tip of the shadow and the base of the tree.
In this case, the angle of elevation is 50 degrees, and the shadow length is 12 meters.
Now, we can use the tangent function, which relates the angle of elevation to the opposite and adjacent sides of a right triangle. In this case, the opposite side is the height of the tree (which we are trying to find), and the adjacent side is the shadow length.
The tangent of an angle is equal to the length of the opposite side divided by the length of the adjacent side. Mathematically, this can be expressed as:
tan(angle) = opposite / adjacent
So, to find the height of the tree, we rearrange the equation:
opposite = tan(angle) * adjacent
Substituting in the given values, we have:
height = tan(50) * 12
Using a scientific calculator or trigonometric table, we can find the tangent of 50 degrees, which is approximately 1.192. Multiplying this by 12 meters, we get:
height ≈ 1.192 * 12
height ≈ 14.3 meters
Therefore, to the nearest tenth of a meter, the height of the pine tree is approximately 14.3 meters.