Question

State the problem-solving strategy described by each of the following. (a) What is the term for writing a series of steps that connects the unknown quantity to a given quantity using unit factors? (b) What is the term for rearranging the variables in an equation in order to solve for the unknown variable?

Short answer

(a) Dimensional Analysis (b) Solving an Equation

Step-by-step solution

Understanding Part (a)

To solve part (a) of the problem, we need to identify the term that describes writing steps to connect an unknown to a known quantity using unit factors. This technique is often used in chemistry, physics, and engineering for converting units.

Identifying the Key Term for Part (a)

The term that refers to this process is 'Dimensional Analysis' or 'Factor Label Method'. It involves multiplying by conversion factors until the desired units are reached. Each conversion is based on unit factors that equate different measurements of the same quantity.

Understanding Part (b)

For part (b), we must find the term that describes rearranging an equation's variables to isolate the unknown variable. This process is fundamental in algebra.

Identifying the Key Term for Part (b)

This process is known as 'Solving an Equation'. It involves algebraic manipulation to isolate the desired variable on one side of the equation to determine its value based on the given quantities.

Key concepts

Dimensional Analysis

Dimensional Analysis is a powerful technique used to convert units from one measurement system to another. It is particularly useful in fields like chemistry, physics, and engineering. The process involves multiplying by conversion factors, which are fractions that represent a relationship between different units of measure. These conversion factors have a value of one, allowing you to change the unit without affecting the magnitude of the physical quantity.

To perform Dimensional Analysis, follow these steps:

• Identify the units you have and the units you need to convert to.
• Set up conversion factors that cancel out the units you want to get rid of.
• Multiply across and ensure all unwanted units cancel out, leaving only the desired units.
• Calculate the final answer after unit cancellation.

This technique ensures accuracy and consistency in scientific calculations involving different measurement systems.

Factor Label Method

The Factor Label Method, which is synonymous with Dimensional Analysis, utilizes conversion factors to change the units of a measured quantity without changing its value. This method is essentially a way to "label" the factors in terms of their units, which aids in keeping track of the conversion process.

For example, if you're given a measurement in meters and need it in kilometers, you would use the factor label method to multiply by a conversion factor (such as \(1 \text{ km} = 1000 \text{ m}\)) to convert the value accordingly. It is imperative in the Factor Label Method that:

• Every step of your conversion should result in the cancellation of the previous unit and the introduction of the desired unit.
• Each conversion factor must be an exact expression of the relationship between units, ensuring that all labels or units correctly represent the equivalency.
• The process requires careful attention to detail to ensure no units are forgotten.

Solving an Equation

Solving an Equation is a fundamental skill in mathematics and science, used to determine the value of an unknown variable. This involves rearranging the equation so the unknown variable stands alone on one side of the equation. Here’s a simplified breakdown of the process:

• Identify the equation and the variable you need to solve for.
• Use algebraic operations such as addition, subtraction, multiplication, and division to isolate the variable. This might involve moving terms from one side to the other.
• Perform the same operation on both sides of the equation to maintain the balance.
• Once the variable is isolated, simplify the expression if necessary to find the value.

Understanding how to solve equations is essential in problem-solving, allowing you to find solutions based on given information. Practicing this skill strengthens your ability to tackle complex problems by breaking them into manageable steps.