Problem 65 - Elementary and Intermediate Algebra

Question

Solve each equation for the specified variable or expression. $$ T=2 \pi \sqrt{\frac{l}{32}} \text { for } l $$

Step-by-Step Solution

Answer

The expression for \( l \) is \( l = \frac{8T^2}{\pi^2} \).

Step 1: Isolate the Square Root

The given equation is \( T = 2 \pi \sqrt{\frac{l}{32}} \). Our goal is to solve for \( l \). Start by isolating the square root term. Divide both sides by \( 2\pi \): \[ \frac{T}{2\pi} = \sqrt{\frac{l}{32}} \]

Step 2: Eliminate the Square Root

To eliminate the square root sign, square both sides of the equation: \[ \left(\frac{T}{2\pi}\right)^2 = \left(\sqrt{\frac{l}{32}}\right)^2 \]Simplifying the right side, we get: \[ \frac{T^2}{4\pi^2} = \frac{l}{32} \]

Step 3: Solve for l

To solve for \( l \), multiply both sides by 32: \[ l = 32 \times \frac{T^2}{4\pi^2} \] Simplify the expression for \( l \) to get the final solution:\[ l = \frac{32T^2}{4\pi^2} \]

Step 4: Simplify the Expression

Simplify \( \frac{32T^2}{4\pi^2} \) by dividing both the numerator and the denominator by 4: \[ l = \frac{8T^2}{\pi^2} \] This is the expression for \( l \) in terms of \( T \).

Key Concepts

Squaring Both Sides

When solving equations that involve a square root, a common technique used is squaring both sides of the equation. This method effectively removes the square root sign and makes the equation easier to solve. Consider the equation we have: \[ T = 2 \pi \sqrt{\frac{l}{32}} \]In step 2 of the solution process, we need to eliminate the square root to further simplify the equation. We do this by squaring both sides:\[ \left(\frac{T}{2\pi}\right)^2 = \left(\sqrt{\frac{l}{32}}\right)^2 \]After squaring both sides, the square root disappears because the operation of squaring and taking a square root are inverse operations. This simplifies our equation to:\[ \frac{T^2}{4\pi^2} = \frac{l}{32} \]Remember, squaring an equation can introduce extraneous solutions, so always check your solutions in the context of the original equation. However, in this problem, since we are dealing with physical quantities, this concern is less prominent.

Isolating Variables

When solving for a specific variable, isolating that variable is one of the first steps. It involves rearranging the equation so that the variable of interest is on one side, often simplifying the path to finding its value. Let's look at our equation again:\[ T = 2 \pi \sqrt{\frac{l}{32}} \]Our task is to solve for \( l \). Initially, isolate the square root term by dividing both sides by \( 2\pi \):\[ \frac{T}{2\pi} = \sqrt{\frac{l}{32}} \]This isolates the square root term itself, which allows further progress towards isolating \( l \). After squaring both sides (as discussed previously), we will need to perform another step to isolate \( l \) completely. The modified equation:\[ \frac{T^2}{4\pi^2} = \frac{l}{32} \]Further, you multiply both sides by 32:\[ l = 32 \times \frac{T^2}{4\pi^2} \]Through careful steps of isolating the terms involving \( l \), we build a pathway to its solution, demonstrating the importance of this fundamental technique.

Simplifying Expressions

After isolating the variable and eliminating unnecessary terms through operations like squaring, it is crucial to simplify the remaining expression to arrive at an accessible and final form. Simplification makes solutions cleaner and often more interpretable.From the previous step, we have:\[ l = 32 \times \frac{T^2}{4\pi^2} \]This expression can be simplified by performing arithmetic operations on the constants: - Divide both 32 and the denominator by 4 to simplify:\[ l = \frac{32T^2}{4\pi^2} \]Simplifying it further gives:\[ l = \frac{8T^2}{\pi^2} \]This reduction not only makes the expression easier to handle but also reveals underlying patterns or relationships in the formula, such as direct proportionality and dependencies on the other variables involved. Thus, simplification is a crucial skill in algebraic manipulation, making derivations clear and solutions more elegant.