Problem 23 - Algebra A Combined Function

Question

If the voltage \(V\) in an electric circuit is held constant, the current \(I\) is inversely proportional to the resistance \(R\). If the current is 40 amperes when the resistance is 270 ohms, find the current when the resistance is 150 ohms.

Step-by-Step Solution

Answer

The current is 72 amperes when the resistance is 150 ohms.

Step 1: Understand the Relationship

The problem describes the relationship between current (\(I\)) and resistance (\(R\)) as inversely proportional, meaning \(I \propto \frac{1}{R}\). This can be expressed as the equation \(I = \frac{k}{R}\), where \(k\) is a constant. Our task is to find \(k\) first.

Step 2: Calculate the Constant (k)

Substitute the known values of current and resistance into the equation. We know \(I = 40\) amperes and \(R = 270\) ohms:\[ 40 = \frac{k}{270} \]To find \(k\), multiply both sides by 270:\[ k = 40 \times 270 = 10800 \]

Step 3: Use k to Find the New Current

Now that we have \(k = 10800\), use this value with the new resistance. Substitute \(R = 150\) ohms into the equation \(I = \frac{k}{R}\):\[ I = \frac{10800}{150} \]Divide 10800 by 150:\[ I = 72 \]

Step 4: Verify the Solution

Verify that when substituting the values back, the formula holds:For \(R = 150\), \(I = \frac{10800}{150} = 72\) amperes. This satisfies the inverse proportionality as expected, verifying the solution is consistent.

Key Concepts

Electric Circuits

Electric circuits are like pathways for electric charges. Imagine them as roads that allow the flow of electricity to travel from one point to another. These circuits can be simple, containing just a battery and a light bulb, or complex, with numerous components like resistors, capacitors, and switches. The essential parts of an electric circuit include:

• Power Source: This could be a battery or a power outlet that provides the electrical energy needed to keep the circuit running.
• Conductors: These are wires or traces on a circuit board that connect different components and allow electrical current to flow.
• Load: Devices like lamps, resistors, or appliances that use the electricity provided by the power source.

If any part of the circuit is broken or if there is a break (like a switch turning off), the current cannot flow, similar to a roadblock on the highway. That's why it's important all parts of a circuit work together harmoniously.

Resistance

Resistance is like a roadblock for electric current. It opposes the flow of electricity, making it harder for current to pass through the circuit. Resistance is measured in ohms (Ω) and is determined by several factors:

• Material: Conductors like copper have low resistance, whereas insulators like rubber have high resistance.
• Length and Thickness: Longer wires have more resistance, while thicker wires have less.
• Temperature: Higher temperatures increase resistance in most conductors.

Understanding resistance is crucial because it affects how much current can flow through a circuit at a given voltage. For example, using a high-resistance bulb in a circuit will result in less bright light since the current flow is reduced.

Current

Imagine current as the traffic of electric charges moving along a circuit. Current, measured in amperes (A), indicates how much charge is flowing through the circuit per second. The more amperes, the more electrical charge is moving.

• Inversely Proportional to Resistance: As seen in the exercise, when voltage is constant, current decreases as resistance increases. This relationship is described by the formula: \( I = \frac{k}{R} \).
• Constant Voltage Effects: A constant voltage means that the only variables affecting current are resistance and the constant multiplier \( k \).

Efficiently managing current is critical for safety and functionality in circuits, ensuring devices can operate without overheating or short-circuiting.

Voltage

Voltage is the driving force that pushes electric current through a circuit. Think of it as the pressure that encourages electrons to move. Voltage is measured in volts (V) and is vital for understanding how circuits function because it dictates the energy available to move charges.

• Constant Voltage Scenario: Like in the given problem, when voltage is constant, the relationship between current and resistance becomes vitally apparent, guiding how they adjust to maintain equilibrium.
• Potential Difference: Voltage is often described as a potential difference, representing the ability to do work or provide energy to move the charge from one point to another.

Without sufficient voltage, current won't flow, and no work gets done in an electric system. That's why batteries and power supplies are so important—without them, circuits wouldn't function.