Problem 38 - Precalculus: Functions and Graphs

Question

\(A\) conveyor belt 9 meters long can be hydraulically rotated up to an angle of \(40^{\circ}\) to unload cargo from airplanes (see the figure). (a) Find, to the nearest degree, the angle through which the conveyor belt should be rotated up to reach a door that is 4 meters above the platform supporting the belt. (b) Approximate the maximum height above the platform that the belt can reach. (IMAGE CAN NOT COPY)

Step-by-Step Solution

Answer

(a) 26°; (b) Maximum height is approximately 5.79 meters.

Step 1: Understanding the Problem

We have a conveyor belt that is 9 meters long and can be raised to function like a ramp at an angle. We need to determine which angle allows the endpoint of the belt to reach a height of 4 meters in the air. Additionally, the problem asks us for the maximum height the belt can reach when raised to its maximum angle of 40 degrees.

Step 2: Apply Trigonometric Concept

We will use trigonometry to solve this problem. The scenario describes a right triangle in which the conveyor belt is the hypotenuse, the height above the platform is the opposite side, and we seek the angle of elevation. Since the opposite side and hypotenuse are involved, we will use the sine function: \[ \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \] In this case: \[ \sin(\theta) = \frac{4}{9} \]

Step 3: Calculate the Angle

We find the angle \( \theta \) by applying the inverse sine (arc sine) function to both sides of the equation: \[ \theta = \sin^{-1}\left(\frac{4}{9}\right) \] Calculating this gives \( \theta \approx 26.38^{\circ} \), which rounds to \( 26^{\circ} \).

Step 4: Maximum Height Calculation

Next, we calculate the maximum height the belt can reach when tilted to its maximum angle of 40 degrees. Using the sine value of the angle: \[ \text{Maximum Height} = 9 \times \sin(40^{\circ}) \] Computing this gives the maximum height as approximately 5.79 meters.

Key Concepts

Right Triangle

In trigonometry, a right triangle is a triangle that has one angle equal to 90 degrees. This type of triangle is fundamental in trigonometry because it allows us to apply various trigonometric functions and relationships to solve problems.

• The longest side of the right triangle is called the hypotenuse.
• The two other sides are known as the opposite and adjacent sides, relative to the angle of interest.

For example, in our conveyor belt problem, the belt forms the hypotenuse of a right triangle when it is raised to unload cargo. Understanding which side is which helps when choosing the appropriate trigonometric function to use in solving for unknown values.

Angle of Elevation

The angle of elevation is the angle formed by the horizontal ground and line of sight to a point above it, such as the top of a building or, in this case, the endpoint of the conveyor belt. When we look up at something at a higher height than where we are, this is the angle we experience.

• In our exercise, we are asked to find the angle of elevation that allows the belt to reach a specific height – 4 meters in this case.
• This angle is crucial to determine how high the conveyor belt needs to be raised to reach the desired height.

Knowing how to calculate this angle using trigonometric functions such as sine aids in engineering tasks and real-world applications where precise movements and measurements are required.

Sine Function

The sine function is one of the primary trigonometric functions and is essential in solving problems involving right triangles. It relates the angle to the ratio of two sides of the right triangle:

• Sine of an angle \( \theta \) is defined as the ratio of the length of the opposite side to the length of the hypotenuse: \( \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \).
• To find an unknown angle, you can use the inverse sine function, often denoted as \( \sin^{-1} \).

In the context of the conveyor belt problem, we used the sine function to determine the angle needed to reach a height of 4 meters when using the belt. By applying the inverse sine function, we could calculate the necessary angle of elevation, which is fundamental in ensuring accuracy in such real-world measurements.