Problem 57 - College Algebra with Corequisite Support

Question

For the following exercises, use the given information to answer the questions. The current in a circuit varies inversely with its resistance measured in ohms. When the current in a circuit is 40 amperes, the resistance is 10 ohms. Find the current if the resistance is 12 ohms.

Step-by-Step Solution

Answer

The current is approximately 33.33 amperes when the resistance is 12 ohms.

Step 1: Identify the Relationship

Since the current varies inversely with its resistance, we can write the relationship between current and resistance using the formula: \( I = \frac{k}{R} \), where \( I \) is the current, \( R \) is the resistance, and \( k \) is a constant.

Step 2: Find the Constant

We have the information that when the current \( I = 40 \) amperes, the resistance \( R = 10 \) ohms. Plug these values into the inverse variation formula to find \( k \): \( 40 = \frac{k}{10} \). Multiply both sides by 10 to solve for \( k \): \( k = 400 \).

Step 3: Use the Constant to Find New Current

Now that we know \( k = 400 \), we can use the formula \( I = \frac{k}{R} \) to find the current when the resistance is \( R = 12 \) ohms. Substitute the known values into the equation: \( I = \frac{400}{12} \). Calculate \( I \) to find that \( I \approx 33.33 \) amperes.

Key Concepts

Current and Resistance

Current and resistance are two fundamental concepts in the study of electricity. They play crucial roles in how electrical circuits function and determine the behavior of electrical components. Current, measured in amperes (A), is the flow of electric charge through a conductor. It tells us how much charge is passing through a point in the circuit every second.

Resistance, measured in ohms (Ω), reflects how much a material opposes the flow of electric current. The greater the resistance, the more difficult it is for current to flow through the conductor. Different materials have different levels of resistance, which impacts how much current can pass through them. Metals, for example, tend to have low resistance compared to materials like rubber, which have high resistance.

• Current indicates the flow of charge through a conductor.
• Resistance opposes current flow and is measured in ohms (Ω).

Understanding the relationship between current and resistance is essential for analyzing and designing circuits.

Ohm's Law

Ohm's Law is a fundamental principle in electronics and electrical engineering, describing the relationship between voltage, current, and resistance in a circuit. According to Ohm's Law, the current \( I \) in a circuit is directly proportional to the voltage \( V \) across the circuit and inversely proportional to the resistance \( R \).The mathematical expression of Ohm’s Law is:

\[ V = I \times R \]

This equation can be rearranged to solve for current or resistance, helping determine one variable when the other two are known. It's an essential tool for designing circuits and diagnosing electrical problems.

For example:

• If the voltage increases while resistance stays constant, the current also increases.
• If the resistance increases while the voltage remains the same, the current decreases.

Understanding Ohm's Law allows you to predict how changes in voltage and resistance affect current flow.

Inverse Proportionality

Inverse proportionality is a type of relationship where one quantity increases as another decreases. In the context of circuits, this means when resistance increases, current decreases, and vice versa. Such relationships are described using the equation:

\[ I = \frac{k}{R} \]

Here, \( I \) is the current, \( R \) is the resistance, and \( k \) is a constant.This equation clearly expresses the inverse relationship between current and resistance.

For practical application, understanding inverse proportionality helps in adjusting circuits for desired performance. For instance, when you increase the resistance in a bulb circuit, the current decreases, dimming the bulb. Conversely, decreasing resistance increases the current, making the bulb brighter.

In solving any problem related to inverse variation:

• Identify the constant of variation, \( k \).
• Use the given data to find missing values by plugging them into the inverse variation formula.

Recognizing inverse proportionality allows an intuitive grasp of how changes in one variable affect another in a tangible way.