Question

At the same time that the sun's rays make a \(60^{\circ}\) angle with the ground, the shadow cast by a flagpole is 24 feet. To the nearest foot, find the height of the flagpole.

Short answer

The flagpole is 42 feet tall.

Step-by-step solution

Understand the Problem

We have a right triangle formed by the flagpole (the height), the shadow on the ground (one leg), and the sunlight angle (the angle above the shadow). The given angle is the angle between the sun's rays and the ground, which is \(60^{\circ}\), and the length of the shadow is 24 feet.

Identify the Relevant Trigonometric Function

Since we have the opposite side (height of the flagpole) to find and the adjacent side (length of the shadow) given, we use the tangent function. In trigonometry, \( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \).

Set Up the Tangent Equation

Using the tangent function, we find that \( \tan(60^{\circ}) = \frac{h}{24} \), where \( h \) is the height of the flagpole.

Solve for the Height

First, calculate \( \tan(60^{\circ}) \), which is \( \sqrt{3} \approx 1.732 \). So, the equation becomes \( 1.732 = \frac{h}{24} \). Multiply both sides by 24 to find \( h \): \( h = 24 \times 1.732 \).

Calculate and Round Off the Height

Perform the calculation: \( h = 24 \times 1.732 = 41.568 \). Since we need the height to the nearest foot, round 41.568 to 42.

Key concepts

Right Triangle

The right triangle is a fundamental concept in trigonometry. It consists of:

• A right angle, which measures exactly 90 degrees.
• Two side legs and a hypotenuse, which is the longest side.

In our exercise, the problem illustrates a right triangle formed by the height of the flagpole, the shadow on the ground, and the line representing the sunlight.
This structure helps us apply different trigonometric functions, such as the tangent function, to solve for unknown sides or angles. In this exercise, the sunlight angle, shadow, and height play critical roles in forming the right triangle used for calculations.

Tangent Function

The tangent function is one of the primary trigonometric functions alongside sine and cosine. It is particularly useful for solving problems involving right triangles.
In terms of a right triangle, the tangent of an angle is defined as the ratio of the length of the opposite side to the length of the adjacent side.

• Mathematically, it is expressed as: \( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \).

In the exercise, by identifying the shadow as the adjacent side and the height as the opposite side, the tangent function allows us to relate these two measurements to the given angle of elevation, making it easier to solve for unknown variables.

Angle of Elevation

The angle of elevation refers to the angle between the horizontal line from the observer's eye to some higher point directly above. In trigonometry, this concept helps define the required angle to solve right triangle problems.
In the problem, the angle of elevation is represented by the 60-degree angle between the sun’s rays and the ground. As the sun is considered higher in the sky, we treat it as an elevated point, with the angle pointing upwards from the shadow to the sun.

• Knowing the angle of elevation helps in setting up equations using trigonometric functions like tangent to find the height in this problem.

Shadow

The shadow in trigonometric problems often serves as a measurable, concrete representation of one side of a right triangle.
In this exercise, the shadow is the horizontal leg of the right triangle formed by the sunlight, flagpole, and the direction of the shadow itself. Here, the shadow measures 24 feet, acting as the adjacent side to the angle of elevation.

• Understanding the role the shadow plays makes it easier to determine the missing dimensions of the triangle, such as the height of the flagpole in this situation.

Height Calculation

Calculating the height of the flagpole using trigonometry begins by applying the tangent function.

• We know the tangent of the 60-degree angle and the length of the shadow.
• Through the equation \( \tan(60^{\circ}) = \frac{h}{24} \), we can solve for \( h \), the height.
• First, calculate \( \tan(60^{\circ}) \) as approximately \( 1.732 \), and then solve the equation \( h = 24 \times 1.732 \).

After performing the multiplication, the resulting value is rounded to the nearest foot for practical use, giving a final height of 42 feet. This way of calculating ensures accuracy while adhering to specified rounding guidelines for real-world application.