Problem 42 - Algebra 2

Question

A transit is 330 \(\mathrm{ft}\) from the base of a building. The angles of elevation of the top and bottom of a flagpole situated on top of the building are \(55^{\circ}\) and \(53^{\circ}\) . Find the height of the flagpole.

Step-by-Step Solution

Answer

The height of the flagpole is \[330ft \cdot (tan(55^{\circ}) - tan(53^{\circ}))\]

Step 1: Calculating the height to the top of the pole

The tangent of an angle in a right-angled triangle is equal to the opposite side divided by the adjacent side. Hence, in the larger right-angled triangle, \[ tan(55^{\circ}) = \frac{height_{top}}{330ft} \] which can be rearranged to give \[ height_{top} = 330ft \cdot tan(55^{\circ}) \]

Step 2: Calculating the height to the bottom of the pole

Similarly, in the smaller right-angled triangle, \[ tan(53^{\circ}) = \frac{height_{bottom}}{330ft} \] which can be rearranged to give \[ height_{bottom} = 330ft \cdot tan(53^{\circ}) \]

Step 3: Calculating the height of the pole

The height of the flagpole is equal to the height to the top minus the height to the bottom. This gives \[ height_{pole} = height_{top} - height_{bottom} = 330ft \cdot (tan(55^{\circ}) - tan(53^{\circ})) \]

Key Concepts

Angles of Elevation

When we talk about the angles of elevation, it's essential to understand what they represent in trigonometry and geometry. An angle of elevation is the angle between the horizontal ground and the line of sight to an object above the observer. This concept is primarily used in real-life applications such as measuring the heights of buildings, mountains, or other elevated points.

• The observer's line of sight follows the line to the top of the object.
• The horizontal line is considered as the ground or the flat baseline.
• The angle is measured from the horizontal up to the line of sight.

By understanding angles of elevation, we can solve many practical problems using trigonometric functions like tangent, which makes calculations easier, especially with inaccessible heights.

Right-Angled Triangles

Right-angled triangles are a cornerstone of trigonometry due to their straightforward properties and the Pythagorean theorem. A right-angled triangle consists of two perpendicular sides and a hypotenuse, which is the side opposite the right angle.

• One angle in the triangle is always 90 degrees.
• The other two angles sum up to 90 degrees.
• Trigonometric functions like sine, cosine, and tangent are often used to relate different sides of these triangles.

In context, when we deal with problems involving right-angled triangles, such as finding the height of a flagpole, we use the properties of this triangle extensively. We usually use a known horizontal distance (adjacent side) and an angle of elevation to find an unknown vertical distance (opposite side). This makes right-angled triangles incredibly useful for such calculations.

Tangent Function

The tangent function is one of the principal trigonometric functions and plays a crucial role in solving right-angled triangle problems, especially those involving angles of elevation or depression.

• The tangent of an angle is defined as the ratio between the opposite side and the adjacent side of a right-angled triangle.
• This is expressed mathematically as \( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \).
• It's particularly useful for finding unknown heights or lengths when an angle and one side are known.

In our exercise, we used the tangent function to determine heights from the ground to both the top and bottom of the flagpole, using the known angle of elevation and horizontal distance. The ability to rearrange the tangent equation to solve for the opposite side is a fundamental trigonometric skill used in various practical applications.