Problem 126 - A Graphical Approach to Precalculus with Limits

Question

Solve each formula for the indicated variable. Leave \(\pm\) in answers when applicable. Assume that no denominators are 0 $$s=s_{0}+g t^{2}+k \quad \text { for } t$$

Step-by-Step Solution

Answer

\(t = \pm \sqrt{\frac{s - s_{0} - k}{g}}\)

Step 1: Isolate the term with t²

Start with the equation \(s = s_{0} + g t^{2} + k\). To isolate the term with \(t^2\), subtract \(s_0\) and \(k\) from both sides to get \(s - s_{0} - k = g t^{2}\).

Step 2: Solve for t²

Now that \(g t^2\) is isolated, divide both sides by \(g\) to solve for \(t^2\). This gives you \(t^2 = \frac{s - s_{0} - k}{g}\).

Step 3: Solve for t

Take the square root of both sides to solve for \(t\). Remember to include the positive and negative solutions due to the square root, resulting in \(t = \pm \sqrt{\frac{s - s_{0} - k}{g}}\).

Key Concepts

Isolation of Variables

The first important step in solving equations, specifically quadratic ones, is the isolation of variables. Here, we focus on isolating variables to make equations easier to solve. Suppose we have an equation like the one given in the exercise: - We started with the equation: \(s = s_{0} + g t^{2} + k\). Our goal was to solve for \(t\). - To isolate \(g t^2\), we first needed to eliminate other terms on the right-hand side. - We did this by subtracting terms \(s_{0}\) and \(k\) from each side of the equation, resulting in: \(s - s_{0} - k = g t^{2}\). By carefully getting rid of other variables and constants, we bring our focus to the needed term. The process involves simple arithmetic operations like subtraction and helps us manage other steps more effectively. Breaking an equation into simpler components is the essence of the isolation process. It's about clearing unnecessary clutter around the variable we are interested in.

Square Roots

Once the desired variable term is isolated and simplified, the next step involves square roots. Square roots are essential when dealing with equations involving squares, such as \(t^2\) in our equation. Let's see how this works: - We had reached a point where \(t^2 = \frac{s - s_{0} - k}{g}\). - The step now was to solve for \(t\). Taking the square root of both sides is necessary here: \(t = \pm \sqrt{\frac{s - s_{0} - k}{g}}\). - Remember, the \(\pm\) symbol indicates there are two possible values. Both positive and negative values are valid for \(t\), as squaring either would return the original square. Understanding square roots is vital in algebra, especially when dealing with quadratic equations. It fully addresses why there can be two solutions for an equation, doubling the importance of each step taken. Square roots aren't just calculations. They're concepts that reveal the symmetry present in equations involving squares.

Formula Manipulation

Formula manipulation involves the strategic rearrangement and solving of equations. It's a dance of operations to express a particular variable. This technique draws on algebraic rules and techniques. Let’s break down the process: - With \(g t^2\) isolated, we divided both sides by \(g\), effectively getting \(t^2 = \frac{s - s_{0} - k}{g}\). This is a classic formula manipulation step to extract \(t^2\) cleanly. - Each operation we perform—subtraction, division, square rooting—involves formula manipulation. It's about applying arithmetic and algebraic operations in a structured, intentional way. - These steps help transform formulas and better understand the relationships between variables. They give us the flexibility to solve for any variable depending on what's needed. Mastering formula manipulation provides the foundation for solving complex equations. It isn't just a skill; it's a toolkit for navigating through mathematical challenges, ensuring precision and confidence in solving for unknowns.