Problem 66 - Algebra for College Students

Question

For the indicated variable. Assume that all letters represent positive numbers. \(s=\frac{1}{2} g t^{2}\) for \(t\)

Step-by-Step Solution

Answer

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

Step 1: Isolate the Fraction

The first step in solving for \( t \) is to eliminate the fraction by multiplying both sides of the equation by \( 2 \). This gives: \[ 2s = g t^2 \]

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

With the fraction removed, divide both sides of the equation by \( g \) to isolate \( t^2 \): \[ t^2 = \frac{2s}{g} \]

Step 3: Solve for \( t \)

Take the square root of both sides to solve for \( t \). The positive square root is used since \( t \) represents time and is positive: \[ t = \sqrt{\frac{2s}{g}} \]

Key Concepts

Solving Equations

Solving equations is like cracking a puzzle. It's about finding the value of the variable that makes the equation true. In our case, the equation is given as \( s = \frac{1}{2} g t^{2} \). Our job is to rearrange this equation to find what \( t \) must be.

The fundamental idea of solving equations is performing operations that simplify the equation while keeping both sides equal. Some basic operations include adding, subtracting, multiplying, or dividing every term on both sides of the equation. If done correctly, you'll end up with the sought variable isolated on one side of the equality symbol. This means you've solved the equation.

Let’s summarize the steps here:

• Identify what you need to solve for.
• Look at the operations involving the variable and undo them step-by-step using inverse operations.
• Keep simplifying until the variable is completely isolated.

Once you have isolated the variable, you've effectively solved the equation for that variable.

Isolating Variables

Isolating variables is a crucial step in solving equations. This process involves manipulating the equation to get the desired variable by itself on one side of the equation. Let's break down how we isolated \( t^2 \) in our example.

Initially, we have \( 2s = g t^2 \) from multiplying both sides by 2 to remove the fraction on the right side of the original equation. Our goal here is to isolate \( t^2 \). To do this, we divide both sides of the equation by \( g \), which means we perform the inverse operation of multiplication (since \( g \) is multiplied by \( t^2 \) in the equation).

Remember:

• Perform the same operation on both sides to keep the equation balanced.
• Use inverse operations to cancel out other terms around your target variable.

After this step, \( t^2 \) is isolated on one side of the equation: \( t^2 = \frac{2s}{g} \). Isolating variables helps simplify the problem, bringing us closer to a solution.

Square Roots

Square roots are essential in solving equations that involve squares of variables. In our solution, the equation is \( t^2 = \frac{2s}{g} \). Since we want to solve for \( t \), we need to find the value of \( t \) that, when squared, equals \( \frac{2s}{g} \).

The square root operation is the inverse of squaring a number. Thus, if we take the square root of both sides of our equation, we will solve for \( t \). This process involves using only the positive square root since time \( t \) cannot be negative.

Here’s a quick guide to remember:

• The square root of \( x^2 \) is \( |x| \).
• When solving equations, consider the context (like positive time) to choose the appropriate root.
• Apply the square root operation equally to both sides of the equation.

For our equation, \( t = \sqrt{\frac{2s}{g}} \) gives us the desired solution. Square roots unlock the ability to retrieve original values from squared expressions.