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Chapter 2: Problem 71

Solve each formula for the specified variable. $$ C=\pi d \text { for } d $$

Short Answer

Expert verified
d = \( \frac{C}{\pi}\)

Step by step solution

01

Understand the Given Formula

The given formula is the circumference of a circle: \(C = \pi d\). The goal is to solve for the variable \(d\).
02

Isolate the Variable

To solve for \(d\), we need to isolate \(d\) on one side of the equation. Currently, \(d\) is multiplied by \(\pi\).
03

Divide Both Sides by \(\pi\)

Divide both sides of the equation by \(\pi\) to isolate \(d\). \[\frac{C}{\pi} = \frac{\pi d}{\pi}\]Simplifying the right-hand side, you get:\[\frac{C}{\pi} = d\]
04

Write the Final Solution

The solution after isolating \(d\) is: \[d = \frac{C}{\pi}\]

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Key Concepts

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

Circumference Formula
The circumference of a circle is the distance around its edge. It's a crucial concept in geometry, especially for measuring circular objects. The formula to find the circumference is:
  • \(C = \pi d\) where \(C\) is the circumference and \(d\) is the diameter of the circle.
Pi (\textrm{\texbecause}) is a constant approximately equal to 3.14159. It's used to relate the diameter to the circumference. By knowing any one of the two variables (\(C\) or \(d\)), you can determine the other using this formula.
Isolating Variables
Often, we need to 'solve for a variable' in a formula, meaning we rearrange the equation to get that variable by itself on one side of the equation. In our problem, we want to isolate \(d\) from the circumference formula \(C = \pi d\).
Here are the steps:
  • Identify the term with the variable you need to isolate (in this case \(d\)).
  • Use mathematical operations to get the variable alone. This can include adding, subtracting, multiplying, or dividing both sides of the equation by the same number.
  • Remember, whatever operation you do to one side of the equation, you must do to the other side to keep it balanced.
These steps help to rearrange the formula to solve for the desired variable.
Algebraic Manipulation
Algebraic manipulation involves rearranging equations to solve for a specific variable. This is crucial in math and science. In this problem, we applied algebraic manipulation to the circumference formula.
  • Start with the formula: \(C = \pi d\)
  • To isolate \(d\), divide both sides by \(\textrm{\texbecause}\): \( \frac{C}{\pi} = d \)
  • Finally, simplify to get: \( d = \frac{C}{\pi} \)

This shows how you can rearrange (manipulate) the given formula to find the variable you're interested in. Mastering these skills makes solving equations easier and more intuitive. Practice is essential in getting comfortable with these techniques.

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