Problem 22 - Precalculus: A Concise Course

Question

The length of a shadow of a tree is 125 feet when the angle of elevation of the sun is \(33^{\circ}\). Approximate the height of the tree.

Step-by-Step Solution

Answer

The height of the tree is approximately 83 feet. Note that this value may vary slightly based on the precision of your calculator's tangent function.

Step 1: Identification of Knowns

Identify the given values from the problem. The length of the shadow (adjacent side) is 125 feet and the angle of elevation (\( \theta \)) is \(33^{\circ}\). We are asked to find the height of the tree (opposite side).

Step 2: Apply the Tangent Function

We can apply the tangent function using the formula \( \tan( \theta ) = \frac{opposite}{adjacent} \). In our scenario, it translates into \( \tan( 33^{\circ} ) = \frac{height}{125} \) . We need to solve this equation for the height of the tree.

Step 3: Solve for the Unknown

Rearranging the above formula to find the height, we get \( height = 125 \times \tan( 33^{\circ} ) \). This will give us the height of the tree.

Key Concepts

Angle of Elevation

The angle of elevation is a vital concept in trigonometry, especially when calculating heights and distances using basic trigonometric functions.
Imagine you are looking up at the top of a tree from a certain point on the ground. The angle your line of sight makes with the ground level is known as the angle of elevation.
This angle helps determine the relationship between the height of the object and its shadow on the ground.

• **It is measured from the horizontal upward to the line of sight.**
• **This angle is often denoted by the symbol \( \theta \).**
• **In practical problems, it helps in calculating the height of tall objects without direct measurement.**

In our given problem, the angle of elevation is \(33^{\circ}\). Understanding and identifying this angle is the first step to solving for the height of the tree.

Tangent Function

The tangent function is one of the primary functions in trigonometry, essential for solving problems involving right triangles.
The function \( \tan(\theta) \) is the ratio of the length of the opposite side to the length of the adjacent side in a right-angled triangle.

• **Mathematically, it is represented as \( \tan(\theta) = \frac{opposite}{adjacent} \).**
• **It is particularly useful when you know one side of a right triangle and an angle, allowing you to find another side.**

In our tree example, the tangent of the angle of elevation (\(33^{\circ}\)) equals the tree's height (opposite side) divided by the shadow's length (adjacent side).
This relationship forms the basis of the equation \( \tan(33^{\circ}) = \frac{height}{125} \). Using this, we rearrange it to find the missing height.

Shadow Length

The shadow length is a crucial component in problems involving the heights of objects and the use of trigonometric functions.

• **When the sun casts a shadow of an object, this shadow acts as the adjacent side in our right triangle model.**
• **It provides a ground-level component used to compute other distances or heights.**

In our exercise, the shadow of the tree, at 125 feet, serves as the key measurement to pair with the tangent function for calculating the tree's height.
Knowing the length of the shadow and utilizing the angle of elevation allows us to create an equation to solve for the unknown height. This is frequently used in real-world applications, like determining building heights or tall structures without direct measurement.