Problem 47 - Precalculus Essentials

Question

A wheelchair ramp is to be built beside the steps to the campus library. Find the angle of elevation of the 23 -foot ramp, to the nearest tenth of a degree, if its final height is 6 feet.

Step-by-Step Solution

Answer

The angle of elevation of the ramp to the nearest tenth is approximately 14.6 degrees.

Step 1: Define the Knowns

Identify the knowns from the problem. The hypotenuse, represented by the wheelchair ramp, is 23 feet. The opposite, represented by the final height, is 6 feet.

Step 2: Apply the Tangent Function

Use the equation for the tangent function. This is \(\tan(θ) = \frac{opposite}{adjacent}\). Assuming the base of the ramp (i.e., the adjacent side) lies flat on the ground, we simplify this equation to \(\tan(θ) = \frac{6}{23}\) given the values of the opposite and hypotenuse sides.

Step 3: Solve for the Angle θ

Rearrange the equation in step 2 to find the angle θ. This can be achieved by applying the inverse tangent function (or \(\arctan\) or \(\tan^{-1}\)) to both sides of the equation, which leads to \(θ = \tan^{-1}(\frac{6}{23})\). Calculate the value of θ using a calculator to the nearest tenth of a degree.

Key Concepts

Angle of Elevation

The angle of elevation refers to the angle between the horizontal ground and an upward line of sight. Imagine looking up from ground level to the top of an object. The angle you raise your eyes through is the angle of elevation. It is an essential concept in geometry and trigonometry. Understanding this angle helps in visualizing how steep a slope is, like the ramp mentioned in the exercise. In practical terms:

• It helps in setting building codes for accessibility, ensuring ramps are safely designed.
• It is used in real-world scenarios like finding the height of tall structures using simple measurements.
• A smaller angle of elevation means a gentler slope, while a larger angle indicates a steeper one.

The angle of elevation also relates to the concept of right triangles in trigonometry, as it always involves the horizontal and a vertical line making an angle with the observer's line of sight.

Tangent Function

The tangent function is one of the basic trigonometric functions and is very useful in solving problems involving right triangles. This function involves the ratio of the length of the side opposite an angle to the length of the side adjacent to that angle.In mathematical terms:

• The tangent of an angle θ is represented as \( an(θ) = \frac{opposite}{adjacent}\).
• It is specifically useful when you know the lengths of opposite and adjacent sides and need to find the angle.

In our exercise, we derived \( an(θ) = \frac{6}{23}\), which emphasized understanding the tangent function.If the angles or sides are not in whole numbers, calculators can quickly solve for tangent ratios with a high degree of accuracy. This functionality makes tangent indispensable for many trigonometric calculations.

Inverse Trigonometric Functions

Inverse trigonometric functions, such as the inverse tangent (\( an^{-1}\) or \(\arctan\)), are vital tools in trigonometry. They let you work backward from known trigonometric values to find an unknown angle measure.When you have a tangent ratio like \(\frac{6}{23}\), the \( an^{-1}\) function can be used to find the angle itself:

• The symbol \(\tan^{-1}\) indicates the inverse tangent.
• Using a calculator, inputting \(\tan^{-1}(\frac{6}{23})\) gives the angle in degrees.

In the exercise, applying this function provided the solution to find the ramp's angle of elevation. Inverse functions are essential for situations where angles are unknown, but side lengths or ratios are. This concept plays a critical role in fields ranging from architecture to engineering, where precise angle measurements are crucial.