Problem 6 - Introductory and Intermediate Algebra for College Students 4th

Question

In Exercises \(1-26,\) solve each formula for the specified variable. Do you recognize the formula? If so, what does it describe? $$C=\pi d \text { for } d$$

Step-by-Step Solution

Answer

The formula for the diameter in terms of the circumference is \(d = \frac{C}{\pi}\).

Step 1: Understand the equation

The formula \(C = \pi d\) gives the circumference of a circle in terms of its diameter. In this equation, \(C\) is the circumference of the circle and \(d\) is the diameter. Our goal is to solve this equation for \(d\). That means making \(d\) the subject of the equation.

Step 2: Solve for the Diameter

To solve for \(d\), we need to isolate \(d\) on one side of the equation. This can be done by dividing both sides of the equation by \(\pi\). This gives us \(d = \frac{C}{\pi}\)

Step 3: Final formula

The formula \(d = \frac{C}{\pi}\) gives the diameter of a circle in terms of its circumference. We have now successfully solved for the diameter.

Key Concepts

Diameter of a Circle

The diameter of a circle is a straight line that passes from one side of the circle to the other, going through the center point. It essentially splits the circle into two equal halves. Understanding the concept of diameter is vital in calculating various properties of a circle. For example:

• The diameter is twice the length of the radius, so if you know the radius ( ) of a circle, you can easily determine the diameter using the formula: \( d = 2r \).
• The diameter directly affects other circle calculations, such as the circumference and the area.

When dealing with the circumference formula \( C = \pi d \), the diameter \( d \) is particularly important because it helps in determining the circle's circumference which is the perimeter or the boundary length of the circle.

Solving for a Variable

When you come across an equation like \( C = \pi d \), solving for a variable means rearranging the equation to find the value of one particular letter, or variable, in terms of others. Here's how you do it for the diameter \( d \):
1. Start with the equation \( C = \pi d \). Here, \( C \) is the circumference, and \(\pi\) is a constant (approximately 3.14159).
2. The goal is to get \( d \) by itself on one side of the equation, meaning we want "\( d = \) something".
3. You achieve this by dividing both sides of the equation by \( \pi \):

• \( d = \frac{C}{\pi} \)

By performing these steps, you've isolated \( d \), giving you a formula to find the diameter if the circumference is known. This is a fundamental aspect of algebra and crucial for solving many real-world mathematical problems.

Circle Formulas

Circle formulas are essential tools in geometry that allow us to calculate various measurements related to a circle.
Some of the most commonly used circle formulas include:

• **Circumference Formula**: \( C = \pi d \) or \( C = 2\pi r \), which allows you to calculate the perimeter of the circle.
• **Area Formula**: \( A = \pi r^2 \), which helps in finding the area within the circle.
• **Diameter from Circumference**: As derived earlier, \( d = \frac{C}{\pi} \), which reverses the circumference calculation to find the diameter.

These formulas apply universally to all circles, regardless of their size. Understanding how and when to use each formula is necessary for solving geometric problems involving circles. Mastering these formulas will also aid in various fields such as engineering, architecture, and even everyday tasks like crafting or decoration.