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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

Identify the formula

The given formula is: \[ C = \pi d \] and you need to solve it for the variable \( d \).
02

Isolate the variable

To isolate \( d \), we need to get \( d \) by itself on one side of the equation. Divide both sides of the equation by \( \pi \) to isolate \( d \): \[ \frac{C}{\pi} = \frac{\pi d}{\pi} \]
03

Simplify

Simplify the right side of the equation: \[ \frac{C}{\pi} = d \] Since \( \pi \) divided by \( \pi \) is 1, the right side simplifies to \( d \). Therefore, the equation becomes: \[ d = \frac{C}{\pi} \]

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Key Concepts

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

Isolating variables
Isolating variables is a key aspect of solving equations. It means you manipulate the equation to get the variable you are solving for alone on one side.
Let's see how we apply this to our problem:
We start with the equation \(C = \pi d\). Here, \(d\) is multiplied by \(\pi\). To isolate \(d\), we divide both sides of the equation by \(\pi\): \[ \frac{C}{\pi} = \frac{\pi d}{\pi} \]\
That leaves us with \(d\) by itself, because \(\pi\) divided by \(\pi\) is 1. Finally, we get \[ d = \frac{C}{\pi} \].
This step-by-step breakdown shows how dividing helps isolate the variable and makes it easier to solve the equation.
Circumference formula
The circumference formula is essential in geometry. It tells us the distance around a circle. The formula is \(C = \pi d\), where \(C\) is the circumference, \(\pi\) is a constant (approximately 3.14159), and \(d\) is the diameter of the circle.
To understand it better:
  • Circumference (C) is the perimeter of the circle, the total distance around it.
  • Diameter (d) is the length across the circle, passing through the center.
  • \(\pi\) is a mathematical constant representing the ratio of the circumference to the diameter of any circle.

Using this formula, we can find the diameter if we know the circumference, as shown in the exercise. Simply rearrange the formula to solve for \(d\).
Algebra steps
Solving equations often involves clear and logical steps in algebra. Here are the key algebra steps used in our exercise:
  • Identify the formula: Start with the given equation \(C = \pi d\).
  • Isolate the variable: Divide both sides by \(\pi\).
  • Simplify: Perform the division to simplify the equation.

These steps ensure we handle equations systematically.
Let's see our exercise again:
Given \[ C = \pi d \], we divided both sides by \(\pi\) to isolate \(d\):
\(\d = \frac{C}{\pi}\) was the result after simplification.
Understanding these algebra steps is crucial because they form the foundation of solving more complex equations in the future.

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