Problem 9 - Elementary and Intermediate Algebra

Question

Tell whether each relationship suggests direct or inverse variation. The force you must exert on the handle of a wrench to loosen a bolt and the length of the handle.

Step-by-Step Solution

Answer

The relationship is an inverse variation.

Step 1: Understanding the Problem

We need to determine whether the relationship between the force exerted on a wrench handle and the length of the handle is direct or inverse variation. In direct variation, two variables change in the same way, whereas in inverse variation, one variable increases while the other decreases.

Step 2: Identifying Variables

Identify the two variables: let the force required to loosen a bolt be represented as \( F \) and the length of the wrench handle as \( L \).

Step 3: Applying the Concept of Lever Arm

Use the concept of torque, which is the product of force and distance from the pivot point. Torque \( T \) is constant for loosening the bolt, so \[ T = F \times L \].

Step 4: Analyzing the Relationship

Given that the torque to loosen the bolt is constant, \( F \) and \( L \) must satisfy the equation \( F \times L = T \). Rewriting, \( F = \frac{T}{L} \). As \( L \) increases, \( F \) decreases proportionally, indicating an inverse variation.

Key Concepts

Direct Variation

In direct variation, one variable changes in the same proportion as another. For students learning this concept, it helps to picture a real-world example, such as the relationship between distance and time when walking at a constant speed. If you walk faster and cover more distance in the same amount of time, both your speed and distance increase together. When two variables like this change in tandem, we say they directly vary with each other. The mathematical representation of direct variation is given by the formula:
\[ y = kx \]
where:

• \( y \) and \( x \) are the variables.
• \( k \) is the constant of proportionality.

This equation indicates that the ratio \( \frac{y}{x} \) remains constant. Direct variation is a powerful tool to predict how changing one variable impacts another, given this constant relationship.

Torque

Torque is an important concept in physics that explains the rotational effect produced by a force. Think of torque as the twist you feel when you turn a doorknob. This rotational force depends on two key factors:

• The amount of force applied.
• The distance from the point the force is applied to the pivot point, known as the lever arm.

The formula for torque is:
\[ T = F \times d \]
where:

• \( T \) is the torque.
• \( F \) is the force applied.
• \( d \) is the lever arm.

It is crucial to note that for a bolt to be loosened, the torque must overcome the resistance holding it in place. Therefore, understanding torque can help in tasks like loosening bolts by using an appropriate application of force over a suitable distance.

Lever Arm

The lever arm is the perpendicular distance between the line of action of the force and the point of rotation, or pivot. It plays a crucial role in determining the effectiveness of the force applied. The longer the lever arm, the less force is required to produce the same amount of torque. This principle explains why wrenches have long handles – to increase the lever arm, allowing the same torque with less force.
When thinking about the mechanics, it's similar to the concept of a seesaw. Placing a child further from the pivot point helps lift a heavier child closer to the center with less effort. Similarly, the equation for torque, \( T = F \times d \), helps us see how increasing the lever arm \( d \) allows for a decrease in the force \( F \) needed, assuming constant torque. Recognizing the role of the lever arm simplifies understanding mechanical advantage in everyday tools and machinery.