Problem 117 - 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 $$s=\frac{1}{2} g t^{2} \quad \text { for } t$$

Step-by-Step Solution

Answer

\(t = \pm\sqrt{\frac{2s}{g}}\).

Step 1: Isolate the Fraction

The given formula is \(s=\frac{1}{2}gt^2\). First, get rid of the fraction by multiplying both sides by 2: \(2s = gt^2\).

Step 2: Solve for \( t^2 \)

Next, to isolate \(t^2\), divide both sides of the equation by \(g\): \(t^2 = \frac{2s}{g}\).

Step 3: Find \(t\) Using Square Root

Take the square root of both sides to solve for \(t\): \(t = \pm\sqrt{\frac{2s}{g}}\). Including \(\pm\) in the solution accounts for both positive and negative roots.

Key Concepts

Quadratic Equations

Quadratic equations involve terms where the variable is raised to the power of two. They can often be written in the standard form: \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants. These types of equations are important because they frequently appear in many areas of math and science, such as physics in calculating projectile motion or determining the roots in algebra.

• If an equation involves a term like \( t^2 \), it's usually a quadratic equation.
• The solutions to quadratic equations can be found using various methods like factoring, completing the square, and the quadratic formula.

In our specific problem, we've dealt with a formula that can be rearranged into the form of solving a quadratic for the variable \( t \). By isolating \( t^2 \), we've essentially been dealing with a quadratic structure, but we took a more direct route to find \( t \) by using square roots.

Variables Isolation

Isolating a variable means to get that variable by itself on one side of the equation. This is an essential skill in algebra for solving an equation and finding the value of the variable. The process can vary depending on the form of the equation.

• Start by simplifying the equation as much as possible.
• Use inverse operations to move other terms to the opposite side. This often involves addition, subtraction, multiplication, or division.
• When dealing with fractions, eliminate the fraction by multiplying both sides of the equation by the denominator.
• In the given problem, we initially multiplied both sides by 2 to eliminate the fraction, focusing on isolating \( t^2 \) by dividing by \( g \).

Simplifying and rearranging equations to isolate variables is key in problem-solving. It allows you to express one variable uniquely in terms of other known or defined quantities.

Square Roots

Understanding square roots is vital when working with quadratic equations, especially when isolating the squared variable. The square root function essentially 'undoes' the squaring of a number.

• The square root of \( x^2 \) is \( x \), but remember this equation typically has two solutions: one positive and one negative (\( x \) and \( -x \)).
• The notation \( \pm \) used in solutions indicates both these possibilities.
• Square roots are often used in physics, engineering, and other sciences to model real-world situations and solve equations derived from those scenarios.

In the context of our problem, taking the square root of both sides was the final step to solve for \( t \). We include the \( \pm \) because both \( t \) and \( -t \) could potentially satisfy the given formula. This is crucial when interpreting solutions to ensure all possibilities are considered.