Problem 8 - Algebra 2

Question

In 1915 , the tallest flagpole in the world was in San Francisco. a. When the angle of elevation of the sun was \(55^{\circ}\) , the length of the shadow cast by this flagpole was 210 ft. Find the height of the flagpole to the nearest foot. b. What was the length of the shadow when the angle of elevation of the sun was \(34^{\circ} ?\) c. What do you need to assume about the flagpole and the shadow to solve these problems? Explain why.

Step-by-Step Solution

Answer

a. The height of the flagpole is approximately 279 feet. b. The length of the shadow when the angle of elevation is \(34^{\circ}\) is approximately 411 feet. c. The assumptions made include that the flagpole is exactly vertical and forms a right angle with the ground, the sunlight strikes the ground at the same angle across the entire flagpole height, and the measurements are accurate.

Step 1: Use Tangent to Solve for Flagpole Height

We can use the tangent of the angle of elevation, which in this case is \(55^{\circ}\). The formula for the tangent of an angle in a right triangle is tangent(angle) = opposite/adjacent. Here, 'opposite' is the height of the flagpole, and 'adjacent' is the shadow length. Rearranging for flagpole height, we get flagpole height = tangent(angle) * shadow length. Plugging in the given values, the flagpole height is approximately \(210 * tan(55^{\circ})\) feet. Using a calculator to evaluate this, the height of the flagpole is about 279 feet.

Step 2: Use the Flagpole Height to Find New Shadow Length

We use the same tangent formula but rearrange for shadow length: shadow length = flagpole height/tangent(angle). Plugging in the calculated height (279 feet) and the new angle of \(34^{\circ}\), the shadow length is approximately \(279 / tan(34^{\circ})\) feet. Using a calculator to evaluate this, we find that the shadow is approximately 411 feet long when the angle of elevation of the sun is \(34^{\circ}\).

Step 3: Discuss Assumptions Made

To solve these problems, we need to assume that the flagpole is perfectly vertical and is a right angle to the level ground, giving us a right triangle with the sun, flagpole tip, and shadow endpoint. We also assume that the sunlight reaches the ground at the same angle throughout the entire height of the flagpole, meaning there are no variances in terrain or other obstacles causing distortions of the shadow. It is also presumed that the measurements given are accurate and there are no measurement errors.

Key Concepts

Tangent function

In trigonometry, the tangent function is one of the basic functions that deals with right triangles. For a given angle in a right triangle, the tangent function calculates the ratio of the opposite side to the adjacent side. This can be expressed as \( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \).
The tangent function is especially useful when you know one side of a right triangle and an angle (other than the right angle).
This formula comes handy in many real-world applications, such as finding the height of tall objects using a shadow and an angle of elevation.

Angle of elevation

The angle of elevation is the angle formed between the horizontal line and the line of sight to an object above the horizontal level. Visualize yourself standing on flat ground, looking up at the top of a tall object. The angle your line of sight makes with the ground is the angle of elevation.
The concept is crucial in problems where you're asked to find the height of an object, like a flagpole, using its shadow on the ground. For instance, if you know the angle of elevation and the shadow length, you can apply trigonometric functions like tangent to calculate the object's height.

Right triangles

A right triangle is a type of triangle that includes one angle measuring exactly 90 degrees. It has several properties that make it a key element in trigonometry.
Knowing just one angle and one side can help you find the remaining sides using trigonometric functions. The sides have specific names:

• The "opposite" side is the one opposite the angle of interest.
• The "adjacent" side is the one next to the angle of interest.
• The "hypotenuse" is the longest side, opposite the right angle.

In shadow problems, right triangles allow us to connect angles, the lengths of shadows, and heights of objects.
This makes calculating dimensions based on angles precise and straightforward.

Shadow problems

Shadow problems are a common type of trigonometric problem that involve estimating the height of an object based on the length of its shadow and the angle of elevation from the ground. These problems use right triangle relationships and trigonometric functions to find unknown values.
To solve a shadow problem, you typically:

• Identify the angle of elevation and the length of the shadow as parts of a right triangle.
• Use trigonometric ratios, like tangent, to find the unknown height of the object casting the shadow.

Assumptions are usually involved in shadow problems, such as the ground being level and the object standing perpendicular to it.
These assumptions aim to keep the right triangle intact, allowing for accurate calculations.