Problem 45 - Elementary and Intermediate Algebra

Question

Solve each problem by writing a variation model. 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 Direct Proportion

The problem states that voltage \( V \) is directly proportional to the current \( I \). This means we can express this relationship as \( V = kI \) where \( k \) is the constant of variation, which in this context is the resistance \( R \). So we modify the equation to \( V = RI \).

Step 2: Substitute Known Values

We are given that the voltage \( V = 6 \) volts and the current \( I = 2 \) amperes. Substitute these values into the equation, so it becomes \( 6 = R \times 2 \).

Step 3: Solve for Resistance

To find the resistance \( R \), we rearrange the equation \( 6 = R \times 2 \) by dividing both sides by 2: \( R = \frac{6}{2} \).

Step 4: Compute the Resistance

Perform the division: \( \frac{6}{2} = 3 \). Therefore, the resistance \( R \) is 3 ohms.

Key Concepts

Variation Model

In mathematics, a variation model is used to describe how one quantity changes when another quantity changes. This is particularly useful when two quantities are directly proportional or inversely proportional to each other. Direct proportionality means that as one variable increases, the other increases at a consistent rate, and this relationship can usually be expressed in a formula like \( y = kx \), where \( k \) is the constant of proportionality.
For the exercise given, we have a direct variation model between voltage \( V \) and current \( I \). The equation representing this relationship is \( V = kI \), where \( k \) is the resistance. The goal is to figure out this constant, which tells us how much resistance a particular resistor has given a certain voltage and current.
Using a variation model allows us to predict one variable based on the known value of another, provided we know the constant of variation.

Voltage and Current Relationship

Electric circuits often operate under the principle that voltage and current are directly proportional. This means that if you increase the current flowing through a circuit, the voltage also increases at a specific rate, assuming the resistance remains constant.
In practical terms, voltage can be considered as the "push" that drives electrons through a conductor, and current as the flow of electrons. The relationship between them can be encapsulated in Ohm's Law, which is a fundamental principle in electronics.

• Ohm's Law states: \( V = IR \)
• \( V \) is the voltage measured in volts
• \( I \) is the current measured in amperes
• \( R \) is the resistance measured in ohms

This means for a given resistor, the voltage across it is determined by the product of the current flowing through it and its resistance.

Constant of Variation

The constant of variation is a key concept when discussing direct proportion. In the context of the problem, this constant is known as resistance, denoted by \( R \). It essentially measures how much the flow of electric current is hindered or resisted by the material of the resistor.
Resistance is measured in ohms and can have a significant impact on how a circuit functions. When working with the equation \( V = RI \), the value of \( R \) helps determine how much voltage is needed to drive a certain amount of current through a resistor.
In our given problem, where \( V = 6 \) volts and \( I = 2 \) amperes, the resistance \( R \) was calculated to be 3 ohms using the rearranged equation \( R = \frac{V}{I} = \frac{6}{2} \). Remember that understanding the constant of variation is crucial for solving electrical circuit problems and predicting how changes in one factor will affect others.