Problem 60 - Elementary Algebra

Question

The breaking distance of an automobile is directly proportional to the square of its speed. The current, \(I\), in an electrical conductor is inversely proportional to its resistance, \(R\). If the current is \(1 / 4\) ampere when the resistance is 100 ohms, then what is the current when the resistance is 150 ohms?

Step-by-Step Solution

Answer

When the resistance is 150 ohms, the current is \( \frac{1}{6} \) ampere.

Step 1: Understanding the Proportionality

The statement tells us that the current, denoted by \( I \), is inversely proportional to the resistance, \( R \). This means we can express this relationship as \( I = \frac{k}{R} \), where \( k \) is a constant of proportionality that we need to determine.

Step 2: Finding the Constant of Proportionality

We are given that the current \( I \) is \( \frac{1}{4} \) ampere when the resistance \( R \) is 100 ohms. Plugging these values into our equation, we have \( \frac{1}{4} = \frac{k}{100} \). To find \( k \), multiply both sides of the equation by 100, resulting in \( k = 25 \).

Step 3: Calculating the Current for 150 Ohms Resistance

Using the constant \( k = 25 \) that we calculated, we can now determine the current when the resistance is 150 ohms. Substitute \( k \) and \( R = 150 \) into the equation \( I = \frac{k}{R} \) to get \( I = \frac{25}{150} \). Simplifying \( \frac{25}{150} \) results in \( I = \frac{1}{6} \) ampere.

Key Concepts

Proportional Relationships

Proportional relationships describe how two quantities change in relation to each other. These can be direct or inverse. In a **direct proportionality**, as one quantity increases, the other also increases at a consistent rate. This is often represented by the formula:

• Direct Proportionality: \( y = kx \)

where \( y \) and \( x \) are the quantities, and \( k \) is the constant of proportionality.

In contrast, an **inverse proportionality** occurs when one quantity increases while the other decreases, maintaining a constant product. It is expressed as:

• Inverse Proportionality: \( y = \frac{k}{x} \)

This is the type of relationship described in the original exercise, where current is inversely proportional to resistance. Understanding these relationships is crucial in mathematics and science, as they allow us to predict how changes in one variable affect another.

Electrical Conductivity

Electrical conductivity is a fundamental concept in physics and engineering. It measures how easily electricity can flow through a material. The principle is reflective of how well a material allows the passage of electrical current.

• A good conductor has high conductivity and allows current to flow easily.
• Poor conductors or insulators have low conductivity and restrict current flow.

In the context of the exercise, conductivity relates to the reciprocal of resistance. The less the resistance, the more conductive a material is, allowing more current to pass through. Conversely, high resistance implies low conductivity and reduces the flow of current.

Conductivity is therefore a convenient way to describe and calculate how efficient a material is at transmitting electrical current, which is crucial in electrical circuits and devices.

Constant of Proportionality

The constant of proportionality is a key component in proportional relationships. It links the varying quantities in a fixed ratio or consistent scaling factor. This constant denotes the strength and type of proportionality present in the equation.

In direct proportionality, the constant dictates how much one variable changes as the other does:

• For example: If \( y = kx \), doubling \( x \) will double \( y \) when \( k \) is constant.

In inverse proportionality, as seen in the original problem, the constant helps maintain the consistent product of the variables, such as:

• If \( I = \frac{k}{R} \), then \( k \) ensures that the product \( IR = k \) is constant, regardless of changes in \( I \) or \( R \).

Finding and understanding this constant allows us to solve and predict outcomes in various real-world scenarios, from chemistry to physics to economics.