Problem 28 - Contemporary Precalculus

Question

A 150 -foot-long ramp connects a ground-level parking lot with the entrance of a building. If the entrance is 8 feet above the ground, what angle does the ramp make with the ground?

Step-by-Step Solution

Answer

Answer: The ramp makes an angle of approximately 3.06° with the ground.

Step 1: Label the given quantities in the right triangle

The 150-foot-long ramp represents the hypotenuse of the right-angled triangle. The leg opposite the angle we want to find is the entrance height, which is 8 feet.

Step 2: Use the sine function to find the angle

Using the sine function, we can set up the equation: sin(angle) = opposite/hypotenuse. Substitute the given values to find the angle. sin(angle) = \frac{8}{150}

Step 3: Find the inverse sine of the equation

To get the angle, take the inverse sine (also known as arcsin or sin^{-1}) of both sides of the equation: angle = sin^{-1}(\frac{8}{150})

Step 4: Calculate the angle

Using a calculator, compute the angle: angle ≈ sin^{-1}(\frac{8}{150}) ≈ 3.06° The ramp makes an angle of approximately 3.06° with the ground.

Key Concepts

Right Triangle

A right triangle is a geometrical shape characterized by having one of its angles measuring exactly 90 degrees. The side opposite to this right angle is known as the hypotenuse, which is the longest side of the triangle. The two other sides are referred to as the legs or catheti. Understanding right triangles is crucial as they form the basis for many concepts in trigonometry.
In problems involving right triangles, you often use the properties of these triangles to calculate angles and side lengths. By applying trigonometric functions like sine, cosine, or tangent, various elements of the triangle can be determined, helping to solve real-world problems like measuring the angle of a ramp or a ladder leaning against a wall.

Sine Function

The sine function is a fundamental concept in trigonometry, commonly used to relate the angles and sides of a right triangle. It is defined as the ratio of the length of the opposite side to the hypotenuse. Mathematically, it is expressed as:

• \( \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \)

Where \(\theta\) is the angle of interest.
In practical scenarios, knowing the sine of an angle helps determine missing dimensions in a right triangle. For instance, in the given exercise, the sine function is used to find the angle that the ramp makes with the ground, given the height of the ramp and its hypotenuse. Overall, the sine function is vital for solving various engineering, physics, and construction problems.

Inverse Sine

The inverse sine function, denoted as \( \sin^{-1} \) or \( \text{arcsin} \), is used to find the angle when the sine value is known. It is essentially the reverse process of the sine function, allowing you to determine the angle from a known ratio of the opposite side to the hypotenuse.
For example, if you know \( \sin(\theta) = x \), then \( \theta = \sin^{-1}(x) \). In the context of the exercise, after finding \( \sin(\theta) = \frac{8}{150} \), we apply the inverse sine to calculate the actual angle:

• \( \theta = \sin^{-1}(\frac{8}{150}) \)
By calculating this, we obtain the angle that the ramp forms with the ground. This process is widely applicable in situations requiring angle determination when particular side lengths are given.

Angle Calculation

Angle calculation in right triangles often involves using trigonometric functions and their inverses. By knowing certain side lengths, like the opposite side and hypotenuse in our example, you can calculate an angle with precision using these mathematical tools.
To solve for an angle, you typically:

• Identify the sides involved - opposite, adjacent, or hypotenuse.
• Select the relevant trigonometric function, usually based on the known sides.
• Apply the inverse function to solve for the angle.

These steps streamline finding an angle, essential for engineering designs, navigation, and various scientific analyses. In the exercise, the calculated angle of approximately 3.06 degrees signifies the ramp's inclination, showcasing the practicality of trigonometry in constructing accessible building entrances. With a calculator, these problems become manageable, providing precise answers to complex scenarios.