Problem 18 - Algebra 2 and Trigonometry

Question

From a point on the ground that is 100 feet from the base of a building, the tangent of the angle of elevation of the top of the building is \(\frac{5}{4} .\) To the nearest foot, how tall is the building?

Step-by-Step Solution

Answer

The building is 125 feet tall.

Step 1: Identify Known Information

We know that the distance from the base of the building to the observation point is 100 feet, and the tangent of the angle of elevation is \(\frac{5}{4}\).

Step 2: Understand Tangent Relationship

The tangent of an angle in a right triangle is the ratio of the opposite side (height of the building) to the adjacent side (distance from the base). Thus, \( \tan(\theta) = \frac{\text{height of building}}{100} = \frac{5}{4} \).

Step 3: Set Up the Equation

Using the tangent relationship, the equation becomes \( \frac{5}{4} = \frac{\text{height of building}}{100} \).

Step 4: Solve for Height

To find the height of the building, rearrange the equation to \( \text{height of building} = 100 \times \frac{5}{4} \).

Step 5: Calculate Height

Perform the multiplication: \( 100 \times \frac{5}{4} = 125 \).

Step 6: Round to Nearest Foot

Since the problem asks for the height to the nearest foot, the calculated height of 125 feet is already a whole number.

Key Concepts

Understanding Tangent in Trigonometry

Tangent is a basic trigonometric function that arises often in problems involving right triangles and angles. It specifically relates the lengths of two sides of a right triangle to each other, based on a particular angle. In a right triangle, tangent is defined as the ratio of the length of the opposite side to the adjacent side, where these sides are referenced from the angle you are considering.
For example, if you have angle \(\theta\) in a right triangle, then \(\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}\). This ratio is powerful because it allows you to connect angular measurements with linear dimensions, which is particularly helpful in real-world height and distance problems.
When given a tangent value, you can often solve for unknown side lengths in right triangles easily using simple algebraic manipulations. Simply put, knowing either side and the tangent of an angle allows you to determine the missing side.

What is Angle of Elevation?

The angle of elevation is an angle formed by the line of sight of an observer looking upward from a horizontal line to an object above the horizontal plane. Imagine standing a certain distance away from a tall building and looking up at the top; the angle your line of sight makes with the horizontally level ground is the angle of elevation.
It's important in trigonometry because it helps in calculating unknown distances and heights using trigonometric functions like tangent.

• The concept comes in handy when we need to measure tall objects without actually being able to climb them, for safety or practicality purposes.
• In exercises, knowing the angle of elevation allows us to use trigonometric ratios to find other unknown measurements, like the height of an object.

Whenever the angle of elevation is involved, always remember that it is measured from the horizontal ground line, moving upwards.

Definition of a Right Triangle

A right triangle is one of the most fundamental shapes in geometry and trigonometry. It is characterized by having one angle exactly equal to 90 degrees. The presence of this right angle gives the triangle its unique properties, and allows the use of trigonometric ratios to find unknown sides and angles.
The right triangle has:

• An opposite side relative to the angle we are analyzing.
• An adjacent side, which is the other side involved in forming the angle but is not the hypotenuse.
• The hypotenuse, the longest side which is always opposite the right angle.

In trigonometry problems, these sides allow for calculations of lengths and angles using functions such as sine, cosine, and tangent. Understanding the layout and properties of right triangles makes applying these trig functions straightforward and practical.

Effective Problem-Solving Steps

Approaching math problems methodically can simplify even complex problems. Let's discuss some effective problem-solving steps using our example.
To solve the problem of calculating the height of a building from a distance, start by:

• Identifying all the given information, such as distances and angles or their trigonometric relations like the tangent.
• Setting up equations using relevant trigonometric identities and solving for the desired unknowns.
• Using algebraic manipulation, such as cross multiplying or direct solving to rearrange equations.
• Perform calculations and verify the result by checking it fits the criteria given in the problem, such as rounding appropriately.

In our building height problem, after setting up the equation from \(\tan\theta\), rearranging allowed us to directly solve for the height as \(125\) feet, verifying the logic that the height calculated made sense based on the triangle's ratio values. This structured approach ensures thoroughness and clarity.