Problem 96 - Intermediate Algebra

Question

Solve each problem by writing a variation model. Electronics. The power (in watts) lost in a resistor (in the form of heat) varies directly as the square of the current (in amperes) passing through it. The constant of proportionality is the resistance (in ohms). What power is lost in a 5 -ohm resistor carrying a 3 -ampere current?

Step-by-Step Solution

Answer

The power lost in the resistor is 45 watts.

Step 1: Understand the Concept of Direct Variation

When a quantity varies directly with the square of another quantity, it means that if the second quantity doubles, the first quantity quadruples, and so forth. In this problem, power varies directly with the square of the current, and the constant of proportionality is the resistance.

Step 2: Write the Direct Variation Model

The problem states that power lost, denoted by \( P \), varies directly as the square of the current, denoted as \( I \). Mathematically, this can be represented as:\[P = R imes I^2\]where \( R \) is the resistance.

Step 3: Substitute Known Values into the Model

The resistance \( R \) is given as 5 ohms, and the current \( I \) is given as 3 amperes. Substitute these values into the variation model:\[P = 5 imes (3)^2\]

Step 4: Solve for Power Lost

Calculate the value by squaring the current and then multiplying by the resistance:\[P = 5 imes 9 = 45\]Thus, the power lost in the resistor is 45 watts.

Key Concepts

Power in Resistors

Resistors, a key component in electrical circuits, dissipate power in the form of heat when current passes through them. This phenomenon is crucial in electronics, as it affects both the performance and efficiency of electrical devices. The power (\( P \)) lost in a resistor is determined by its resistance (\( R \)), which is a measure of how much the resistor opposes the flow of electric current.

The formula used to calculate the power dissipated in a resistor is derived from the relationship between power, current, and resistance:

• Power is directly proportional to the square of the current (\( I^2 \)).
• The constant of proportionality in this case is the resistance (\( R \)).

Thus, we have the formula \( P = R \times I^2 \), where increasing the current through the resistor leads to a quadratic increase in power dissipation.

Understanding how power is lost in resistors can help in designing circuits that are safe and efficient, preventing components from overheating and ensuring longevity and reliability.

Ohm's Law

Ohm's Law is a fundamental principle in electronics and physics. It reveals the relationship between voltage (\( V \)), current (\( I \)), and resistance (\( R \)). This law is pivotal when analyzing electrical circuits and voltages across components.

The mathematical representation of Ohm's Law is:

• \( V = I \times R \)

This formula signifies that the voltage across a resistor is the product of the current flowing through it and its resistance.

For example, if you have a circuit with a resistance of 5 ohms and a current of 3 amperes, as stated in the problem you solved, you can use Ohm’s Law to find the voltage:

• \( V = 3 \times 5 = 15 \) volts

Ohm's Law helps in understanding how adjustments in voltage, current, or resistance can affect a circuit, which is essential for troubleshooting and optimizing electronic circuits.

Mathematical Modeling

Mathematical modeling involves creating equations to represent real-world situations, making it easier to predict outcomes and analyze complex systems. In electronics, such models can elucidate relationships between different physical parameters, like the one established between power, current, and resistance in resistors.

The exercise you completed is an example of a direct variation model. Here, you modeled the relationship using a simple equation: \( P = R \times I^2 \).

In this model:

• The power (\( P \)) lost in the resistor is the dependent variable.
• The square of the current (\( I^2 \)) acts as the independent variable affected by its magnitude.
• The resistance (\( R \)) is the constant of proportionality facilitating the calculation.

Building such models allows us to predict how changing one variable, like the current, will impact other variables within the system. This predictive power makes mathematical modeling invaluable, especially in engineering and technology fields where understanding and predicting system behavior is crucial.