Problem 50 - Algebra and Trigonometry

Question

A 20-ft ladder is leaning against a building. If the base of the ladder is 6 ft from the base of the building, what is the angle of elevation of the ladder? How high does the ladder reach on the building?

Step-by-Step Solution

Answer

The ladder reaches a height of approximately 19.08 ft, and the angle of elevation is about 72.54°.

Step 1: Understanding the Problem

We have a right triangle formed by the ladder, the wall, and the ground. The ladder is the hypotenuse, which is 20 ft. The horizontal distance from the wall to the base of the ladder is 6 ft. We need to find the angle of elevation and the height at which the ladder touches the wall.

Step 2: Identify the Right Triangle Components

In the right triangle, the ladder itself is the hypotenuse ( 20 ft), the base is the adjacent side (6 ft), and the height of the ladder reaching the building is the opposite side.

Step 3: Use the Pythagorean Theorem

We use the Pythagorean theorem to find the height of the ladder on the building. The formula is: \[a^2 + b^2 = c^2\]where \(a = 6\) ft, \(c = 20\) ft, and \(b\) is the height. Solving for \(b\):\[6^2 + b^2 = 20^2\]\[36 + b^2 = 400\]\[b^2 = 364\]\[b = \sqrt{364}\]\[b \approx 19.08 \, \text{ft}\]

Step 4: Calculate the Angle of Elevation

To find the angle of elevation \(\theta\), use the cosine function, since we have the adjacent side (6 ft) and the hypotenuse (20 ft). So,\[\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{6}{20}\]\[\cos(\theta) = 0.3\]Now find \(\theta\) by taking the inverse cosine:\[\theta = \cos^{-1}(0.3) \approx 72.54^\circ\]

Key Concepts

Right Triangle

A right triangle is a type of triangle where one of the angles is exactly 90 degrees. This distinct feature makes right triangles incredibly useful in trigonometry. They allow us to work with straightforward calculations involving the lengths of sides and angles.
The right triangle is composed of:

• The hypotenuse, which is the longest side opposite the right angle.
• The adjacent side, which is the side connected to the angle of interest besides the hypotenuse.
• The opposite side, which is the side opposite the angle of interest.

Every scenario involving right triangles can rely on these definitions to solve problems like finding unknown side lengths or angles.

Pythagorean Theorem

The Pythagorean theorem is a crucial mathematical principle in dealing with right triangles. It relates the lengths of the sides in the triangle and is expressed by the equation: a^2 + b^2 = c^2 This theorem tells us that the square of the hypotenuse ( c ) is equal to the sum of the squares of the other two sides ( a and b ).
When applying the Pythagorean theorem to find an unknown side, ensure you:

• Identify the hypotenuse correctly.
• Plug in known side values.
• Solve for the unknown side using algebraic manipulation.

In the given problem, we discovered the ladder touches the building approximately 19.08 feet high using this theorem.

Angle of Elevation

The angle of elevation is a real-world application of trigonometry. It is the angle above the horizontal line from an observer's view, upwards to some point of interest.
To calculate it accurately in a problem:

• Identify the sides involved: adjacent and hypotenuse.
• Use trigonometric ratios to solve for the angle.

In our scenario, the angle of elevation is the angle formed between the ground (the horizontal line) and the ladder (the line of sight). Using known sides, you can determine this angle accurately, which was approximately 72.54 degrees.

Cosine Function

The cosine function is an essential trigonometric function represented as:\( \\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} \)This ratio is valuable when solving for angles or sides in a right triangle.
To find the angle:

• First, determine the adjacent side and the hypotenuse.
• Calculate the ratio and use the inverse cosine function to find the angle.

In our exercise, using the cosine function allowed us to find the angle of elevation, proving its usefulness in exercises involving right triangles.