Problem 36 - Precalculus: Functions and Graphs

Question

A builder wishes to construct a ramp 24 feet long that rises to a height of 5.0 feet above level ground. Approximate the angle that the ramp should make with the horizontal.

Step-by-Step Solution

Answer

The angle the ramp should make with the horizontal is approximately 12.0 degrees.

Step 1: Identify the Right Triangle

The ramp forms a right triangle with the ground, where the ramp length is the hypotenuse, the vertical rise is the opposite side, and the horizontal base is the adjacent side.

Step 2: Understand Trigonometric Ratios

To find the angle that the ramp makes with the horizontal, we use the sine trigonometric ratio. For a right triangle, \( \sin(\theta) \) is defined as the ratio of the length of the opposite side to the hypotenuse. Here, that is \( \sin(\theta) = \frac{5.0}{24} \).

Step 3: Solve for the Angle

Use the inverse sine function to find \( \theta \). Calculate \( \theta = \sin^{-1} \left( \frac{5.0}{24} \right) \).

Step 4: Calculate the Angle

Using a calculator, compute \( \theta \approx \sin^{-1}(0.2083) \). This results in \( \theta \approx 12.0 \) degrees.

Key Concepts

Right Triangle

In the world of geometry, a right triangle is a triangle with one of its angles measuring precisely 90 degrees. This creates a unique relationship among the triangle's sides. The side opposite to the right angle is known as the hypotenuse, which is also the longest side. The other two sides are referred to as the legs of the triangle. In practical scenarios, right triangles often appear in constructions, like ramps or incline planes.

When building a ramp, the structure typically forms a right triangle where:

• The ramp itself is the hypotenuse.
• The vertical rise from the ground is one leg (opposite the angle of interest).
• The horizontal distance along the ground forms the other leg.

This configuration allows us to apply trigonometry for calculating angles and lengths, which is essential in architectural design.

Sine Function

The sine function is a fundamental concept in trigonometry, associated with angles in right triangles. It's commonly referred to when dealing with height and distance problems, like our ramp example. The sine of an angle in a right triangle is defined as the ratio of the length of the side opposite the angle to the hypotenuse.

Mathematically, this is expressed as:\[\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}\]
In our ramp problem, this translates to:

• \(\text{Opposite Side} = 5.0 \text{ feet}\)
• \(\text{Hypotenuse} = 24 \text{ feet}\)

So, the sine function helps us determine the angle with ease, using these lengths.

Inverse Sine

Inverse trigonometric functions are used to find angles when the values of the trigonometric ratios are known. The inverse sine, known as \(\sin^{-1}\) or arcsin, is specifically used to calculate the angle when the sine value is given.

For our ramp scenario, where we have the sine as:\[\sin(\theta) = 0.2083\]We calculate the angle \(\theta\) by finding the inverse sine:\[\theta = \sin^{-1}(0.2083)\]
This function is crucial in converting the known ratio back into an angle, making it a key tool in various engineering and physics calculations.

Angle Calculation

To determine the angle of inclination of the ramp, you use the combination of the sine function and its inverse. Starting with our known values from the sine ratio, we employ a calculator to evaluate the inverse sine.

Given:\[\theta = \sin^{-1}(0.2083)\]This helpful operation concludes by estimating \(\theta\) to be approximately 12 degrees. Using a calculator simplifies this process significantly, ensuring you have a quick and accurate result for the angle.

The precise calculation of this angle is critical in designing ramps to ensure they are accessible and safe for use, reflecting how integral trigonometry is in practical applications.