Question

To guard against a fall, a ladder should make an angle of \(75^{\circ}\) or less with the ground. What is the maximum height that a 20-foot ladder can reach safely?

Short answer

The maximum height is approximately 19.3 feet.

Step-by-step solution

Understand the Problem

We need to find the maximum height a 20-foot ladder can reach when the angle between the ladder and the ground is at most 75 degrees. We will use trigonometry since we have a right-angled triangle made by the ladder (hypotenuse), the ground (adjacent side), and the height (opposite side).

Identify the Relevant Trigonometric Function

Since we want to find the height (opposite side), and we know the ladder's length (hypotenuse) and the angle, we will use the sine function. Recall that \( \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \).

Set Up the Equation

Using the sine function: \( \sin(75^{\circ}) = \frac{\text{height}}{20} \). We need to solve this for the height.

Calculate \( \sin(75^{\circ}) \)

Use a calculator to find that \( \sin(75^{\circ}) \approx 0.9659 \).

Solve for Height

Substitute \( \sin(75^{\circ}) \approx 0.9659 \) into the equation: \( 0.9659 = \frac{\text{height}}{20} \). Multiplying both sides by 20 gives \( \text{height} = 20 \times 0.9659 \).

Final Calculation and Conclusion

Perform the multiplication: \( \text{height} = 19.318 \). Therefore, the maximum safe height the ladder can reach is approximately 19.3 feet.

Key concepts

Understanding Right-Angled Triangles

A right-angled triangle is a type of triangle that includes a 90-degree angle. This is crucial in various branches of mathematics, especially in trigonometry, because it allows us to identify and understand the relationships between the triangle's sides and angles.
In the context of a ladder problem, imagine the situation as a right-angled triangle:

• The ladder represents the hypotenuse (the longest side of the triangle).


• The height the ladder reaches against the wall is the opposite side.


• The ground on which the ladder rests is the adjacent side.

These components form the basis for using trigonometric functions to solve problems involving right-angled triangles. Understanding these relationships helps in using the correct function and solving for unknown variables effectively.

Exploring the Sine Function

The sine function is one of the fundamental trigonometric functions used extensively in geometry to relate angles and side lengths in right-angled triangles. In general, for a given angle \( \theta \) in a right-angled triangle:

• The sine of \( \theta \) is the ratio of the length of the opposite side to the hypotenuse.


• This is expressed in the formula: \( \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \).

For our ladder problem, the sine function helps calculate the maximum height when the angle is 75 degrees. Knowing the hypotenuse (the ladder's length) and wanting to find the opposite side (the height), we simply rearrange the formula to solve for the height. The sine function makes it possible to relate the angle directly to the physical height the ladder will reach, enhancing both safety and accuracy.

Angles in Geometry and Their Significance

Angles play a crucial role in geometry, serving as the cornerstone for understanding shapes and their properties. In any triangle, the angles determine the kinds of trigonometric relationships one can use.

• A specific angle like 75 degrees, as in our ladder problem, instructs which trigonometric function is applicable and how it relates the sides of the triangle.


• Understanding the properties of angles also aids in visualizing and solving geometrical problems intuitively.

It's notable that angles not only help determine the dimensions of a right-angled triangle but also aid in ensuring safety, like keeping a ladder angle shallow enough (75 degrees or less) to prevent a fall. This safety angle ensures the ladder remains stable and reaches an optimal height efficiently.