# A student stands 20 m away from the foot

## Question

A student stands 20 m away from the foot of a tree and observes that the angle of elevation of the top of the tree, measured from a table 1.5 m above the ground, is 34°28′. Calculate the height of the tree to the nearest metre.

The height of the tree to the nearest meter is 15 m.

### Explanation

To find the height of the tree, we need to break down the problem into two main components:

1. The height from the table to the top of the tree (h1).
2. The height of the table above the ground (h2).

Data:

• Distance from the student to the foot of the tree = 20 m
• The height of the table above the ground, h2 = 1.5 m
• The angle of elevation from the table to the top of the tree, θ = 34°28′

The next step is to use trigonometry to find the height from the table to the top of the tree. We need to remember that the angle of elevation forms a right triangle with the tree height (from the table to the top) as the opposite side and the distance from the student to the foot of the tree as the adjacent side.

Hence, we need to convert the angle from degrees and minutes to decimal form:

34°28′ = 34 + (28/60) = 34.4667°

The adjacent = Distance from the student to the foot of the tree

In this case: Tan(34.4667°) = opposite / 20

Therefore, Solving for the height from the table to the top of the tree: Height from the table to the top of the tree = 20 x Tan⁡(34.4667°)

We can calculate this using a calculator: Tan ⁡(34.4667°) ≈ 0.6864

Therefore, height from the table to the top of the tree = 20 x 0.6864

Now, we add the height of the table to this value to find the total height of the tree: Total height of the tree = 13.728 m + 1.5 m ≈ 15.228 m

Therefore, To the nearest metre, the height of the tree is: 15 meters