Question

Paul wants to determine the height, x, of a nearby tree. He stands 47 feet from the base of the tree. The measure of the angle of elevation from Paul to the top of the tree is 36º.

Select all of the following that can be used to find the height of the tree?

Answer 1

The following can be used to find the height of the tree:

1. Trigonometry, specifically the tangent function.

2. The Pythagorean Theorem.

Step-by-step explanation:

To determine the height of the tree, x, Paul can utilize trigonometry and the Pythagorean Theorem. The angle of elevation, 36º, is crucial for these calculations.

Firstly, the tangent function relates the angle of elevation
`(\(\theta\))` to the opposite side (height of the tree, x) and the adjacent side (distance from the base, 47 feet). Using the formula
`\(\tan(\theta) = (x)/(47)\)` , Paul can solve for x.

Secondly, the Pythagorean Theorem can be applied. Considering the right triangle formed by Paul, the tree, and the ground, the theorem states that the square of the hypotenuse (47 feet) is equal to the sum of the squares of the other two sides (height of the tree, x, and the distance from the tree to Paul on the ground). This can be expressed as
`\(47^2 = x^2 + d^2\)`, where d is the distance on the ground. Solving for x will give the height of the tree.

In conclusion, trigonometry and the Pythagorean Theorem provide effective methods for Paul to determine the height of the nearby tree.