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Chapter 1: Problem 6

Solve each formula for the specified variable. \(d=r t\) for \(t \quad(\) distance \()\)

Short Answer

Expert verified
The solution for \(t\) is \(t = \frac{d}{r}\).

Step by step solution

01

Understand the formula

The formula given is for distance: \[d = r t\] where: \[d\] = distance \[r\] = rate or speed \[t\] = time
02

Identify the variable to solve for

We need to solve the formula for the variable \(t\) (time).
03

Isolate the variable \(t\)

To solve for \(t\), we can isolate it by dividing both sides of the equation by \(r\): \[d = r t\] Divide both sides by \(r\): \[\frac{d}{r} = t\]
04

Write the final answer

After isolating \(t\), we get: \[t = \frac{d}{r}\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Distance Formula
The distance formula is a fundamental concept in algebra and physics. It is expressed as \(d = r t\), where:
  • \textbf{d}: Distance traveled
  • \textbf{r}: Rate or speed
  • \textbf{t}: Time taken
This formula tells us that the distance covered is equal to the rate at which you're traveling, multiplied by the time spent traveling. Understanding this formula can help in solving various practical problems, like figuring out how long it will take to reach a destination if you know your speed.
Variable Isolation
Variable isolation is a crucial algebraic skill. It involves rearranging an equation to solve for a particular variable. In our problem, we are isolating \(t\), which represents time. Here are the steps to isolate a variable:
  • Identify the variable you need to isolate.
  • Perform algebraic operations to get that variable by itself on one side of the equation.
Let's isolate \(t\) in \(d = r t\):
1. Identify the variable (in this case \(t\)).
2. Divide both sides by the rate of speed (\textbf{r}) to get: \[ \frac{d}{r} = t \] The variable \(t\) is now isolated. You've solved for time!
Algebraic Manipulation
Algebraic manipulation is the process of using algebraic rules and operations to simplify, solve, or rearrange equations. This includes operations like addition, subtraction, multiplication, and division applied to both sides of the equation.
In the given problem, we performed algebraic manipulation to isolate the variable \(t\) by dividing both sides of the equation \(d = r t\) by \(r\). Here鈥檚 a quick recap:
  • Start with the equation: \(d = r t\)
  • Divide both sides by \(r\): \[ \frac{d}{r} = t \]
  • Simplify to get the isolated variable: \[ t = \frac{d}{r} \]
Through these steps, you learn to manipulate the equation to highlight the desired variable. This skill is invaluable in solving a wide range of algebraic problems.

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