Question

MODELING WITH MATHEMATICS The Duquesne Incline in Pittsburgh, Pennsylvania, has an angle of elevation of 30°. The track has a length of about 800 feet. Find the height of the incline.

Short answer

The height of the incline is 400 feet.

Step-by-step solution

Understanding the problem

We have a right triangle formed by the incline, the horizontal ground and the height of the incline. The angle of elevation is given as 30° and the hypotenuse (the incline itself) is 800 feet. We need to find the opposite side of the triangle, which is the height of the incline.

Apply the sine function

In a right triangle, the sine of an angle is defined as the length of the opposite side divided by the length of the hypotenuse. In this case, \(\sin(30°) = \frac{height}{800}\). Therefore, \(height = 800 \cdot \sin(30°)\).

Solve for the height

Now we just need to plug in the value of \(\sin(30°)\), which is 0.5. Thus, \(height = 800 \cdot 0.5 = 400\) feet.

Key concepts

Angle of Elevation

When you hear the term "angle of elevation," think about looking up from a horizontal line. It's the angle formed from the ground up to the top of an object. Imagine standing at the base of a hill and looking upward towards the peak. The angle your line of sight makes with the ground is the angle of elevation.
In trigonometry, this concept helps us to solve real-world problems involving right triangles. For instance, it allows us to determine the height of tall objects by knowing the distance from the object and the angle of elevation.
When working with these problems, always remember:

• Identify the location of the angle in the right triangle.
• The angle of elevation is always measured from the horizontal plane.
• It's crucial for finding unknown distances using trigonometric functions like sine, cosine, or tangent.

Sine Function

The sine function is one of the primary trigonometric functions used to relate the angles and sides of right triangles.
In a right triangle, the sine of an angle is the ratio of the length of the side opposite the angle to the length of the hypotenuse. It is expressed as:
\[ \sin(\theta) = \frac{\text{opposite side}}{\text{hypotenuse}} \]
This ratio enables us to calculate unknown distances or angles when certain lengths are known. In our incline problem:

• The angle of elevation is 30°
• The hypotenuse (incline length) is 800 feet

Therefore, the height (opposite side) can be found as:\[ \text{height} = 800 \times \sin(30°) = 800 \times 0.5 = 400 \text{ feet} \]
Sine values for common angles like 30°, 45°, and 60° are often memorized or readily available in trigonometry tables to simplify calculations in such problems.

Right Triangle Properties

Right triangles are fundamental in geometry and trigonometry because they have distinct properties that make calculations simpler.
A right triangle consists of a 90° angle and two other acute angles. The side opposite the right angle is the longest and called the hypotenuse. The other two sides are known as the adjacent and opposite sides relative to the angle in question.
Key properties include:

• The Pythagorean theorem, which relates the squares of the sides: \( a^2 + b^2 = c^2 \) where \(c\) is the hypotenuse.
• Trigonometric ratios such as sine, cosine, and tangent that link angles to side lengths.
• Right triangles can be used in various applications, including engineering and architecture, to calculate heights and distances.

By understanding these properties, you can solve problems involving unknown sides or angles effectively, as we did in the incline problem by using the sine function.