Problem 38 - Algebra and Trigonometry with Analytic Geometry

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.

Step-by-Step Solution

Answer

(a) 26°, (b) Approximately 5.79 meters.

Step 1: Understanding Right Triangle Formulation

We are given a right triangle formed by the conveyor belt as the hypotenuse, the height from the platform to the airplane door as one leg, and the base along the platform as the other leg. We need to find the angle of rotation for the conveyor belt that makes its opposite side (height from platform to door) 4 meters.

Step 2: Using Trigonometric Ratio to Find the Angle

The sine function relates the angle of rotation θ to the opposite side (4 meters) and hypotenuse (9 meters) of the triangle. The equation is \( \sin(\theta) = \frac{4}{9} \). To find \( \theta \), take the inverse sine: \( \theta = \sin^{-1}\left(\frac{4}{9}\right) \).

Step 3: Calculating the Angle

Compute \( \theta = \sin^{-1}\left(\frac{4}{9}\right) \), which gives approximately \( \theta \approx 26.4^{\circ} \). Rounding this to the nearest degree gives \( 26^{\circ} \).

Step 4: Verifying the Angle

Verify by recalculating sine for \( 26^{\circ} \) to ensure \( \sin(26^{\circ}) \approx 0.444 \), which closely matches \( \frac{4}{9} \approx 0.444 \), so the angle is correct.

Step 5: Maximum Height Calculation

To find the maximum height, we set the conveyor belt at the maximum angle of \(40^{\circ}\). Use the sine function: \( \sin(40^{\circ}) = \frac{\text{maximum height}}{9} \) to get maximum height \( = 9 \cdot \sin(40^{\circ}) \approx 5.79 \) meters.

Key Concepts

Right Triangle Problems

Right triangle problems are common in scenarios involving angles and lengths, such as conveyor belts or ladders. A right triangle consists of:

• Hypotenuse: The longest side, opposite the right angle.
• Adjacent side: The side next to the angle in question.
• Opposite side: The side opposite the angle in question.

When dealing with the conveyor belt problem, it's essential to visualize the belt forming the hypotenuse of a right triangle. The task is to find an angle such that the opposite side is 4 meters. To start, identify the known quantities:

• Length of the conveyor belt (hypotenuse): 9 meters.
• Height from the platform to the door (opposite side): 4 meters.

These setups guide us in selecting the appropriate trigonometric function to solve for the angle.

Inverse Trigonometric Functions

Inverse trigonometric functions are critical in solving for angles when two sides of a triangle are known. For the conveyor belt problem, we use the inverse sine function, or arcsine, because of the known opposite and hypotenuse sides.The inverse sine function, denoted as \( \sin^{-1}(x) \), finds the angle \( \theta \) whose sine is \( x \) (i.e., \( \sin(\theta) = x \)). In our exercise, we have:\[ \theta = \sin^{-1}\left(\frac{4}{9}\right) \]Solving this gives \( \theta \approx 26.4^{\circ} \). This angle represents the rotation needed for the belt to reach the door at 4 meters height. Always ensure to round to the nearest degree in practical applications, resulting in \( 26^{\circ} \). This value can be verified by taking the sine of the calculated angle to ensure accuracy.

Sine Function Calculations

The sine function is fundamental in determining either angles or lengths in right triangles. It is particularly useful when dealing with heights and angles as seen in the conveyor belt problem.To calculate maximum height reach, we first recognize the sine function as it relates an angle \( \theta \) to the opposite side over the hypotenuse:\[ \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \]Given the maximum rotation angle of \(40^{\circ}\), we calculate:\[ \sin(40^{\circ}) = \frac{\text{max height}}{9} \]Solving for the maximum height:\[ \text{max height} = 9 \cdot \sin(40^{\circ}) \]This results in a maximum height of approximately \(5.79\) meters. Understanding how to manipulate the sine function enables determining other triangle sides and is valuable in various engineering applications.