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Chapter 11: Problem 5

Solve the formula for the given variable. $$d=r t ; t \quad \text { (Physics) }$$

Short Answer

Expert verified
The formula, when solved for time (t), is \(t = \frac{d}{r}\).

Step by step solution

01

Identifying the given formula

The problem includes the formula \(d = rt\). This formula is often used in physics, it indicates that distance is equal to rate times time.
02

Transforming the formula

To solve for the variable \(t\), we need to isolate it. To do this, we divide both sides of the equation by \(r\). This will cancel out \(r\) on the right side of the equation, leaving \(t\) by itself.
03

Performing the operation

By applying the operation, we get the new formula : \(t = \frac{d}{r}\). which is the formula solving for \(t\).

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Key Concepts

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

Distance-Rate-Time Problem
In the context of physics and everyday life, understanding the relationship between distance, rate, and time is crucial. This relationship is often encapsulated in the distance-rate-time formula, expressed as:
\[ d = rt \]
This formula helps us calculate one of the variables when the other two are known. Here:
  • \(d\) represents distance, which can be the total length covered by an object or a person.
  • \(r\) stands for rate, commonly referred to as speed. It indicates how fast the object is moving.
  • \(t\) is time, the duration during which the movement occurs.
Understanding how to manipulate this formula is not just a matter for physics but also helps in planning travel schedules and estimating arrival times. By rearranging the formula, you can solve for any variable, depending on the given information.
Algebraic Manipulation
Algebraic manipulation is the process of rearranging equations and expressions to solve for a particular variable or to simplify expressions. It is a fundamental skill in mathematics. In the distance-rate-time equation, algebraic manipulation helps us express time \(t\) in terms of distance \(d\) and rate \(r\). This involves:
  • Identifying the part of the equation to isolate. For instance, if we need to solve for \(t\), we focus on isolating this variable.
  • Applying mathematical operations that preserve the equality. In our example, both sides of the equation are divided by \(r\) to solve for \(t\).
After performing these steps, the original equation \(d = rt\) is transformed into \(t = \frac{d}{r}\). This form directly provides us with the time when distance and rate are known. Mastering algebraic manipulation enables you to tackle various mathematical problems efficiently.
Isolating Variables
Isolating a variable involves rearranging an equation so that the variable of interest stands alone on one side. This is especially useful when you need direct expressions to calculate or understand specific aspects of a problem.
To isolate a variable:
  • Identify the operations currently affecting the variable. These operations must be systematically reversed to separate the variable.
  • Conduct inverse operations to strip away these influences. In our distance-rate-time equation, we divided by \(r\) because it was the operation linking \(t\) to \(d\).
  • Perform these steps carefully on both sides of the equation to maintain balance.
By isolating the variable \(t\) in the formula \(t = \frac{d}{r}\), it becomes clearer to directly determine the time based on distance and rate values. This method can help solve more complex problems by using simpler arithmetic or algebraic techniques.

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