karen, (who is 1,6m tall) wishes to calculate the height of a tall tree. she stands 50m from the base of the tree and measures its angle of elevation to be 36 degrees.
determine the height of the tree?
review your basic trig definitions, and draw a diagram. You will see that
(h-1.6)/50 = tan 36°
Now just solve for h.
To determine the height of the tree, Karen can use trigonometry, specifically the tangent function. The tangent of an angle can be used to find the ratio of the opposite side to the adjacent side in a right triangle. In this case, the adjacent side is the distance from Karen to the base of the tree and the opposite side is the height of the tree.
First, let's label the sides of the triangle:
- The opposite side is the height of the tree (h).
- The adjacent side is the distance from Karen to the base of the tree (d).
Given:
- Distance from Karen to the base of the tree (d) = 50m.
- Angle of elevation (θ) = 36 degrees.
To determine the height of the tree (h):
1. Convert the angle from degrees to radians by multiplying it by π/180:
θ_radians = 36 * π/180
2. Now we can use the tangent function to find the height of the tree:
tan(θ_radians) = h / d
Rearranging the equation to solve for h:
h = d * tan(θ_radians)
Plugging in the values given:
h = 50 * tan(36 * π/180)
3. Use a calculator to find the value of tan(36 * π/180) and evaluate it:
tan(36 * π/180) ≈ 0.7265
Calculate the height of the tree:
h = 50 * 0.7265
Therefore, the height of the tree is approximately:
h ≈ 36.325 meters.