Problem 95 - Intermediate Algebra

Question

Solve each problem by writing a variation model. Electronics. The voltage (in volts) measured across a resistor is directly proportional to the current (in amperes) flowing through the resistor. The constant of variation is the resistance (in ohms). If 6 volts is measured across a resistor carrying a current of 2 amperes, find the resistance.

Step-by-Step Solution

Answer

The resistance is 3 ohms.

Step 1: Understand the relationship

The problem states that voltage is directly proportional to the current flowing through the resistor. This means we can express this relationship as \( V = kI \) where \( V \) is the voltage in volts, \( I \) is the current in amperes, and \( k \) is the constant of variation, which in this case is the resistance \( R \).

Step 2: Identify known values

We are given that the voltage \( V = 6 \) volts and the current \( I = 2 \) amperes. We need to find the resistance \( R \), which is the same as the constant \( k \) in the direct proportionality equation.

Step 3: Set up the equation

Using the direct variation model \( V = kI \), substitute the known values: \( 6 = k \times 2 \).

Step 4: Solve for the resistance

Rearrange the equation to solve for \( k \): \[ k = \frac{6}{2} \].This simplifies to \( k = 3 \).

Step 5: Interpret the result

The resistance, \( R \), is the constant \( k \) we solved for. Thus, the resistance is 3 ohms.

Key Concepts

Direct Proportionality

Direct proportionality is a concept where two quantities increase or decrease together at the same rate. When one quantity doubles, the other also doubles, and so on. This can be mathematically expressed as \( y = kx \), where \( y \) and \( x \) are the two quantities, and \( k \) is the proportionality constant. In this formula, \( k \) remains the same regardless of the specific values of \( y \) and \( x \).

For example, consider the situation discussed in our problem involving

• voltage (\( V \)) as directly proportional to current (\( I \)), and
• resistance (\( R \)), which is the constant of proportionality.

Understanding that the relationship \( V = kI \) forms a direct proportionality line means that if you plot these relationships on a graph, it will be a straight line through the origin.
This clear, predictable relationship simplifies our calculations considerably and is foundational in electronics and many other physics-related fields.

Voltage and Current Relationship

The relationship between voltage and current is a cornerstone of electronics and can be described through Ohm’s Law. Ohm's Law states that the voltage across a resistor is directly proportional to the current flowing through it. When one changes, the other changes in a consistent manner that is determined by the resistor’s resistance.

The core equation derived from Ohm’s Law is

• \( V = IR \), where
• \( V \) is the voltage measured in volts,
• \( I \) is the current measured in amperes, and
• \( R \) is the resistance measured in ohms.

This means that if you know any two of these values, you can find the third one easily. For instance, in our problem, when given

• \( V = 6 \) volts and \( I = 2 \) amperes,

you can substitute these values into the formula to find the resistance.
This direct relationship aids not only in solving mathematical problems but also in understanding how electronic circuits behave under different conditions.

Resistance Calculation

Resistance is a measure of how much a component resists the flow of electric current, and it's crucial for determining how much current will flow for a given voltage. Calculating resistance involves rearranging the fundamental formula derived from Ohm’s Law: \( R = \frac{V}{I} \).

This formula tells us that resistance is the ratio of voltage to current.

In the solved exercise, knowing

• \( V = 6 \) volts and \( I = 2 \) amperes, the calculation for resistance becomes straightforward as:
• \( R = \frac{6}{2} = 3 \) ohms.

Upon solving this, you find that the resistance is 3 ohms, signifying how much the current flow is opposed at that voltage level.
This measurement is fundamental for designing and understanding circuits, ensuring that devices operate safely and effectively by controlling current flow.