Problem 57 - College Algebra

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 \(12 \mathrm{ohms}\).

Step-by-Step Solution

Answer

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

Step 1: Understand Inverse Variation

In inverse variation, if two variables, say \(I\) (current) and \(R\) (resistance), vary inversely, it implies that \(I \times R = k\), where \(k\) is a constant. For this problem, it is given that current and resistance vary inversely.

Step 2: Find the Constant of Variation

Use the given values to find the constant \(k\). When the current \(I = 40\) amperes, and the resistance \(R = 10\) ohms, we use the inverse variation formula. \[ I \times R = k \] \[ 40 \times 10 = k \] Therefore, \(k = 400\).

Step 3: Use the Constant to Find the New Current

Now that we have \(k = 400\), use it to find the current when the resistance is 12 ohms. \[ I \times 12 = 400 \] Solve for \(I\): \[ I = \frac{400}{12} \] \[ I = rac{100}{3} \approx 33.33 \]

Key Concepts

Current and Resistance

In electrical circuits, current and resistance share a unique inverse relationship. This means that as the resistance increases in a circuit, the current decreases, and vice versa. This inverse relationship is vital in understanding how electrical circuits function.

• Current (I): Measured in amperes (A), it represents the flow of electric charge.
• Resistance (R): Measured in ohms (Ω), it signifies how much a circuit resists the flow of current.

When dealing with changes in these variables, knowing this inverse relationship allows us to predict how one will change when the other is altered. For example, if the resistance in a circuit is increased, assuming all other factors remain constant, the current will decrease. This principle underlies many practical applications, such as adjusting the resistance of a circuit to control current flow and prevent overheating.

Constant of Variation

In problems involving inverse variation, the constant of variation (k) is crucial. It quantifies the specific relationship between the variables in a given situation. Inverse variation is expressed as:\[ I \times R = k \]Where:

• \( I \) is the current
• \( R \) is the resistance
• \( k \) is the constant of variation

To find \( k \), you multiply the current by the resistance when both values are known. Once the constant is determined, it can be used to solve for unknown values of current or resistance under new conditions. In our example:\[ k = 40 \times 10 = 400 \]This constant tells us that any product of current and resistance in this context must equal 400. This consistence aids in calculating unknowns efficiently.

Ohm's Law

Ohm's Law is a fundamental principle in electrical engineering. Although distinct from inverse variation, it complements understanding of current and resistance relationships.Ohm's Law is represented by the formula:\[ V = I \times R \]Where:

• \( V \) is the voltage across the circuit
• \( I \) is the current
• \( R \) is the resistance

The law states that the voltage (V) across a circuit is proportional to the product of the current (I) and the resistance (R). This means that if you know any two of these three variables, you can calculate the third.While Ohm's Law involves direct relationships rather than inverse ones, it is helpful in creating a more complete picture of circuit behavior. It's often used alongside concepts of inverse variation to analyze and design circuits.