Problem 9 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Some kids are sitting on their stoop wondering about the height of a tall street post. They estimate that the street post is casting a shadow 15 feet in length and that the angle of elevation of the sun (from the ground) is about \(60^{\circ}\). Estimate the height of the street post.

Step-by-Step Solution

Answer

The height of the street post is approximately 26 feet.

Step 1: Understand the Given Information

The problem tells us that there is a right triangle made up of the street post (which we're trying to find the height of), its shadow, and the angle of elevation of the sun. We know the length of the shadow is 15 feet and the angle of elevation is \(60^{\circ}\). The street post is opposite to the angle of elevation and the shadow forms the adjacent side of the triangle.

Step 2: Apply the Tangent Rule

In a right triangle, the tangent of an angle is equal to the ratio of the length of the side opposite to it to the length of the side adjacent to it. We can write this in the equation form as \(\tan(\Theta) = \frac{\text{Opposite side}}{\text{Adjacent side}}\).

Step 3: Solve for the Height of the Street Post

We substitute the given values into the equation. So, \(\tan(60^{\circ}) = \frac{\text{height of the street post}}{15 feet}\). After solving it, the height of the street post will be \(15 feet * \tan(60^{\circ})\).

Key Concepts

Understanding a Right Triangle

A right triangle is a triangle where one of the angles is exactly 90 degrees, making it a central component in various trigonometric calculations. In this scenario, the right triangle is formed by the street post, the shadow it casts, and the line of sight to the sun's angle of elevation.
The three sides of a right triangle are:

• The hypotenuse: this is the longest side and is opposite the right angle.
• The opposite side: in this problem, it's the vertical street post whose height we seek to measure.
• The adjacent side: this is the shadow on the ground, which is given as 15 feet.

Right triangles are incredibly useful in practical problems, like measuring heights and distances, due to their predictable geometry and the trigonometric functions that arise from their structure.

Exploring the Tangent Function

The tangent function is one of six main trigonometric functions. It relates the angles of a right triangle to the ratio of its sides. Specifically, the tangent of an angle in a right triangle is the ratio of the length of the side opposite the angle to the length of the adjacent side.
Mathematically, it is expressed as:

• \( \tan(\Theta) = \frac{\text{Opposite side}}{\text{Adjacent side}} \)

In this specific exercise, the tangent function helps calculate the height of the street post by utilizing the angle of elevation (\(60^{\circ}\)). By knowing the shadow's length (adjacent side) and using the tangent function, you can determine the height of the post (opposite side).
This approach simplifies finding dimensions in real-world contexts where measuring directly is difficult or impossible.

Grasping the Angle of Elevation

The angle of elevation is the angle formed by the line of sight when looking upward from a lower point to a higher object. In our street post problem, it's the angle from the end of the shadow on the ground looking up to the top of the post.
The angle of elevation plays a crucial role in problems involving height and distance because:

• It provides a clear path to apply trigonometric ratios, like the tangent function, helping to solve unknown heights or distances.
• It is measured from the horizontal upward, making it straightforward to apply within right triangles where horizontal and vertical components are clear.

By understanding the angle of elevation, we can work with trigonometric functions to determine unknown measurements based on provided angles and distances, offering a practical solution for problems in geometry and surveying.