Problem 83 - Beginning and Intermediate Algebra

Question

Solve. \(w=\sqrt{\frac{1}{L C}}\) for \(C\) (An electricity formula)

Step-by-Step Solution

Answer

Isolating C, we have: \(C = \frac{1}{w^2L}\)

Step 1: Square both sides

To get rid of the square root, square both sides of the equation: \((w)^2 = \left(\sqrt{\frac{1}{LC}}\right)^2\)

Step 2: Simplify the equation

After squaring both sides, we have: \(w^2 = \frac{1}{LC}\)

Step 3: Invert both sides

Now, we'll invert both sides of the equation to get the C in the numerator: \(\frac{1}{w^2} = LC\)

Step 4: Isolate C

We want to get C by itself, so we'll divide both sides by L: \(\frac{1}{w^2L} = C\) This is the final solution, with C isolated and expressed in terms of w and L: \(C = \frac{1}{w^2L}\)

Key Concepts

Electrical Circuits

Electrical circuits are fundamental to understanding how electrical components work together to perform a function. An electrical circuit serves as a pathway that allows electricity to flow from one point to another and can include components like resistors, capacitors, and inductors.
In terms of our problem, the formula \(w = \sqrt{\frac{1}{LC}}\) is related to the resonance frequency of a circuit. Understanding this formula is crucial in designing circuits for technologies like radios and TVs because it helps predict how circuits will behave at different frequencies.
It’s essential to grasp how elements like inductance \(L\) and capacitance \(C\) balance each other to find the resonance point and optimize the output of the circuit. This is why isolating a variable within a formula might be necessary, adjusting circuit parameters as needed.

Solving Equations

Solving equations is a fundamental skill in algebra and is crucial for finding unknown values within mathematical expressions. In our example, solving for \(C\) within the formula \(w = \sqrt{\frac{1}{LC}}\) involves manipulating the equation to make \(C\) the subject.
This process requires a systematic approach to maintain the equation's balance, often involving operations such as squaring, inverting, multiplying, and dividing both sides of the equation. Knowing how to effectively navigate each step ensures accuracy and helps prevent errors.
By understanding this process, students can solve real-world problems in various fields, not only in electrical circuits but also in physics and engineering, where equations are used to model and predict complex systems.

Isolation of Variables

The isolation of variables involves rearranging an equation so that a specific variable is by itself on one side of the equation. This skill is important in algebra because it allows us to solve for unknowns and understand the relationships between different components of a formula.
In the context of the given problem, isolating \(C\) from the formula \(w = \sqrt{\frac{1}{LC}}\) involves several steps: squaring to remove the square root, inverting to bring \(C\) to the top, and finally dividing to separate \(C\) from \(L\).
Mastering this technique helps students break down problems into manageable steps, making it easier to address more complex equations. Problem-solving becomes more intuitive and less daunting as students become familiar with each operation's purpose and effect.