Problem 92 - A Graphical Approach to Precalculus with Limits

Question

A force of 30 pounds is required to hold an 80 -pound pressure washer on an incline. What angle does the incline make with the horizontal?

Step-by-Step Solution

Answer

The incline makes an angle of approximately 22.02 degrees with the horizontal.

Step 1: Identify the Forces Involved

In this problem, we are dealing with a weight of 80 pounds and a force of 30 pounds acting to hold it in place on an inclined plane. The weight acts downwards directly, while the holding force acts parallel to the incline.

Step 2: Understand the Relationship Between Forces and Angle

On an inclined plane, the weight of an object can be resolved into two components: one perpendicular to the plane and another parallel to the plane. The parallel component, which is responsible for the object sliding down the incline, is given by \(W \sin(\theta)\), where \(\theta\) is the angle of the incline and \(W\) is the weight of the object.

Step 3: Set Up the Equation

To hold the object in place, the holding force must counteract the parallel component of the weight. Therefore, the equation becomes:\( F = W \sin(\theta) \).Here, \(F = 30 \text{ pounds}\) and \(W = 80 \text{ pounds}\). Substitute the given values into the equation:\( 30 = 80 \sin(\theta) \).

Step 4: Solve for \(\theta\)

Rearrange the equation to solve for \(\sin(\theta)\):\( \sin(\theta) = \frac{30}{80} = 0.375 \).Now, use the inverse sine function to find \(\theta\):\( \theta = \arcsin(0.375) \).

Step 5: Calculate the Angle

Using a calculator, find \(\theta\):\( \theta \approx 22.02^\circ \).

Key Concepts

Inclined Plane

An inclined plane is a flat surface that is tilted at an angle to the horizontal. It's one of the simplest machines and is used to make it easier to move heavy objects up or down. This is because the inclined plane reduces the amount of force needed by increasing the distance over which the force is applied. For example, a ramp is an inclined plane that enables you to lift something heavy over a greater distance using less force. In trigonometry and physics, the inclined plane problem often involves calculating the forces needed to hold or move an object up the incline. Understanding the forces or angles involved helps in making real-life tasks, like loading a truck with heavy equipment, more manageable. Just as in our problem, where a force is used to hold a pressure washer on the angled surface without it sliding back down.

Component Forces

When dealing with inclined planes, you must understand how forces can be broken down into horizontal and vertical components. These are known as component forces. Any force applied on a slope can have a portion that acts parallel to the slope and another that acts perpendicular.- **Parallel Force**: This component tends to slide the object downwards along the plane. For a weight, it's calculated as \[ W \sin(\theta) \], where \( W \) is the weight and \( \theta \) is the angle of the incline.- **Perpendicular Force**: This force acts directly against the plane and helps maintain contact with the inclined surface.In the exercise, the 30-pound force acts parallel to the inclined plane to counteract the sliding force of the washer's weight. Understanding these components is key to solving similar problems efficiently.

Inverse Sine Function

The inverse sine function, also called arcsine, is crucial in trigonometry for finding angles when the sine of the angle is known. It is symbolized as \( \arcsin(x) \) and is the opposite of the sine function.Suppose you know the ratio of the lengths of the opposite side to the hypotenuse in a right triangle. In that case, the inverse sine function helps you find the angle itself. In our exercise, we set up the equation based on the forces:\[ \sin(\theta) = 0.375 \]To find \( \theta \), you use the inverse sine function:\[ \theta = \arcsin(0.375) \]This step is essential in trigonometry when solving for angles, especially in problems involving inclines and component forces.

Angle Calculation

Calculating the angle of an incline involves systematically setting up an equation from known values and then solving that equation. It requires an understanding of trigonometric functions to relate the sides of a triangle to its angles.In problems involving inclined planes, once you've resolved the weight into its components, you use equations like\[ F = W \sin(\theta) \]to find the desired angle \( \theta \). After substituting the known values, you rearrange to solve for the sine function:\[ \sin(\theta) = \frac{30}{80} = 0.375 \]Then, inverse trigonometric functions such as \( \arcsin \) are used to determine the actual angle:\( \theta \approx 22.02^\circ \)This approach is fundamental in trigonometry and is used frequently to calculate angles in various applications, making it indispensable in physics and engineering.