To find the height of the tree, we will use trigonometry, specifically the tangent function, which relates the angle of elevation with the opposite side (the height of the tree) and the adjacent side (the distance from the bottom of the tree).
We're given:
- The angle of elevation is 54°.
- The distance from the tree on the ground ("adjacent side" in the context of the right triangle we are considering) is 52 feet.
We want to find:
- The height of the tree ("opposite side" of the right triangle).
Here's how we can calculate it:
1. Write down the relationship given by the tangent function: tan(θ) = opposite/adjacent.
2. Substitute the known values into the equation:
tan(54°) = height_of_tree / 52 feet.
3. Solve for the height_of_tree:
height_of_tree = tan(54°) * 52 feet.
To proceed with the calculations, we will need to find the value of tan(54°). We use a scientific calculator or a trigonometry table for that purpose. Typically, you would need to find the tangent of 54° in decimal form, which is approximately 1.3764.
Once we have the value for tan(54°), we can substitute it into the equation:
height_of_tree = 1.3764 * 52 feet.
Now we perform the multiplication:
height_of_tree = 71.5728 feet.
As the final step, we round the value to the nearest hundredth as instructed:
height_of_tree ≈ 71.57 feet.
So, the tree is approximately 71.57 feet tall.
