Problem 87 - A Graphical Approach to Precalculus with Limits

Question

A force of 25 pounds is required to hold an 80 -pound crate from rolling on a hill. What angle does the hill make with the horizontal?

Step-by-Step Solution

Answer

The hill makes an angle of approximately 18.21° with the horizontal.

Step 1: Understanding the Problem

A force of 25 pounds is being used to prevent the crate from rolling down the hill. This means that 25 pounds of force are used to counteract the component of gravitational force pulling the crate down the slope. We need to find the angle \( \theta \) that the hill makes with the horizontal.

Step 2: Setting Up the Problem

The force pulling the crate down the slope is the component of the gravity acting parallel to the slope. This component can be given by \( F_{ ext{parallel}} = W \cdot \sin(\theta) \), where \( W \) is the weight of the crate, and \( \theta \) is the angle of the hill.

Step 3: Applying Trigonometric Functions

Since the force required to stop the crate from rolling is given to be 25 pounds, we set: \[ F_{ ext{parallel}} = W \cdot \sin(\theta) = 25 \]We know \( W = 80 \) pounds, so the equation becomes:\[ 80 \cdot \sin(\theta) = 25 \]

Step 4: Solving for \( \sin(\theta) \)

Solve the equation for \( \sin(\theta) \) by dividing both sides by 80:\[ \sin(\theta) = \frac{25}{80} \]

Step 5: Calculating the Angle \( \theta \)

Use the inverse sine function to find \( \theta \):\[ \theta = \arcsin\left(\frac{25}{80}\right) \]

Step 6: Final Computation

Using a calculator, compute the inverse sine:\[ \theta \approx \arcsin(0.3125) \approx 18.21^\circ \]

Key Concepts

Inverse Sine Function

The inverse sine function, often denoted as \( \arcsin \) or \( \sin^{-1} \), is a trigonometric function used to determine an angle from a given sine value.
This function essentially "reverses" the sine, allowing us to find the angle \( \theta \) when we know \( \sin(\theta) \).

In the exercise, once we have the equation \( \sin(\theta) = 0.3125 \), the inverse sine function helps solve for \( \theta \).

• The primary range of \( \theta \) for \( \arcsin \) is between \(-90^\circ\) and \(90^\circ\), which are the valid outputs for \( \arcsin \).
• Computing \( \arcsin(0.3125) \) means we find the specific angle whose sine value is 0.3125.

For practical applications, particularly in physics and engineering, using \( \arcsin \) is essential when working backwards from known measurements, such as in this problem where we deduced the angle from a known force and weight.

Gravitational Force

Gravitational force is one of the fundamental forces of nature and is responsible for attracting two bodies toward each other.
It is described by the formula \( F = mg \), where \( m \) is the mass of an object and \( g \) is the acceleration due to gravity (approximately 9.8 m/s² on Earth).

In the context of this exercise, we're dealing with a component of gravitational force.

• The crate has a weight of 80 pounds, which is essentially the gravitational force acting on it.
• To keep the crate from rolling, a 25-pound force is needed to counter the component of this gravitational force that acts parallel to the slope.

By understanding how the gravitational pull is split into components, especially the one parallel to a surface, we see how it affects movement and the forces required to maintain stability.

Angle of Inclination

The angle of inclination refers to the tilt of a surface relative to the horizontal plane. This angle, often denoted as \( \theta \), plays a crucial role in various real-world applications like construction, physics, and especially in slope stability.

• In this problem, the angle of inclination of the hill affects how much force is required to prevent an object from rolling down.
• A steeper angle might require more force to counteract gravity's pull.

To determine this angle from a real-life scenario, trigonometric functions and their inverses are typically used, as seen with the \( \arcsin \) function in the solution.
This calculation helps in designing systems or understanding phenomena, like why trucks might have difficulty climbing steep hills.