Problem 37 - Precalculus Mathematics for Calculus

Question

A 20 -ft ladder is leaning against a building. If the base of the ladder is \(6 \mathrm{ft}\) 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 angle of elevation is approximately \(72.54^\circ\), and the ladder reaches approximately 19.08 ft up the building.

Step 1: Set Up the Problem

Let's identify the parts of the problem: we have a right triangle formed by the ladder, the building, and the ground. The ladder serves as the hypotenuse of the right triangle, the distance from the building is one leg (6 ft), and the height the ladder reaches up the building is the other leg. We need to find the angle of elevation (angle between the ground and the ladder) and the height reached by the ladder.

Step 2: Use the Pythagorean Theorem

To find the height (h) the ladder reaches on the building, use the Pythagorean theorem: \( a^2 + b^2 = c^2 \). Here, \( a = 6 \) ft, \( c = 20 \) ft, and \( b = h \).Substitute the known values:\[6^2 + h^2 = 20^2\]\[ 36 + h^2 = 400\]Subtract 36 from both sides:\[h^2 = 364 \]Take the square root:\[ h = \sqrt{364} \approx 19.08 \text{ ft}\]

Step 3: Find the Angle of Elevation

Use the trigonometric ratio: \( \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} \), which is suitable because we know the base of the building (adjacent side) and the ladder (hypotenuse).\[ \cos(\theta) = \frac{6}{20} \]\[ \cos(\theta) = 0.3 \]Now, calculate the angle \( \theta \):\[ \theta = \cos^{-1}(0.3) \approx 72.54^\circ \]

Key Concepts

Pythagorean theorem

The Pythagorean theorem is your best friend when dealing with right triangles. It's a fundamental concept in trigonometry that allows us to find the length of any side of a right triangle as long as we know the lengths of the other two sides. The theorem is defined by the formula, \(a^2 + b^2 = c^2\), where \(c\) is the hypotenuse and \(a\) and \(b\) are the other two sides.

In the exercise given, our ladder represents the hypotenuse \(c\), standing at 20 ft. The distance from the building represents side \(a\), or 6 ft. To find \(b\), the height the ladder reaches up on the building, we substitute these values into our formula to get \(6^2 + h^2 = 20^2\). After solving for \(h\), we find the height the ladder reaches is about 19.08 ft.

This formula is a core concept in finding unknown side lengths in right-angled triangles, making it extremely useful in various practical applications like construction, navigating slopes, and more.

angle of elevation

The angle of elevation denotes the angle between the horizontal ground and a line of sight to a point above. It's crucial in various fields, such as engineering and architecture, to determine how objects at different elevations relate to their base points. In our scenario, we are calculating how steep the ladder is leaning against the building.

For the angle of elevation, we utilize trigonometric ratios, such as cosine, which involves the adjacent side (the distance on the ground) and the hypotenuse (the length of the ladder). The relationship is given by \(\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}\).

• Here, the adjacent side is 6 ft long.
• The hypotenuse is 20 ft.
• Thus, \(\cos(\theta) = \frac{6}{20} = 0.3\).

To find the angle \(\theta\), the equation \(\theta = \cos^{-1}(0.3)\) gives us \(\theta \approx 72.54^\circ\). This tells us the ladder leans sharply to the building, making it quite upright.

right triangle

The right triangle is a fundamental element in trigonometry, characterized by having one 90-degree angle. Understanding its properties is crucial to solving many real-life problems. Right triangles consist of three sides: the hypotenuse (the longest side), the opposite side (opposite the angle of interest), and the adjacent side (next to the angle of interest).

In this context, the ladder forms the hypotenuse of the right triangle based on its position against the building and the ground. The opposite side is the height reached by the ladder, while the adjacent side is the distance from the building base to the ladder base.

• In general, the hypotenuse is always opposite the 90-degree angle.
• The nature of a right triangle allows the application of the Pythagorean theorem and trigonometric ratios effectively.

When dealing with these triangles, identifying the sides relative to the angle of interest is key to correctly applying trigonometric functions and solving for unknown quantities in given problems.