Problem 53 - College Algebra with Corequisite Support

Question

For the following exercises, solve for the given variable in the formula. After obtaining a new version of the formula, you will use it to solve a question. The formula for the circumference of a circle is \(C=2 \pi r .\) Find the circumference of a circle with a diameter of 12 in. (diameter = \(2 r)\). Use the symbol \(\pi\) in your final answer.

Step-by-Step Solution

Answer

The circumference is \( 12 \pi \) inches.

Step 1: Understand the Formula

The formula given is for the circumference of a circle: \( C = 2 \pi r \). Here, \( C \) is the circumference and \( r \) is the radius of the circle.

Step 2: Relate Diameter to Radius

Since we are given the diameter, we need to relate it to the radius. The diameter \( d \) is twice the radius, so \( d = 2r \), which implies \( r = \frac{d}{2} \).

Step 3: Find the Radius

The exercise gives the diameter as 12 inches. Using the relationship \( r = \frac{d}{2} \), we find \( r = \frac{12}{2} = 6 \) inches.

Step 4: Substitute Radius into Formula

Substitute \( r = 6 \) into the circumference formula: \( C = 2 \pi r = 2 \pi \times 6 \).

Step 5: Calculate the Circumference

Perform the multiplication: \( 2 \times 6 = 12 \). Therefore, \( C = 12 \pi \).

Step 6: Final Answer

The circumference of the circle with a diameter of 12 inches is \( 12 \pi \) inches.

Key Concepts

Circle Geometry

Circle geometry is fundamental to understanding the properties and calculations related to circles. A circle is a simple shape in plane geometry consisting of all points in a plane that are at a certain distance, called the radius, from a fixed point known as the center. Key attributes within circle geometry include:

• **Radius:** The distance from the center of the circle to any point on the circle.
• **Diameter:** The longest distance across the circle, passing through the center, which is twice the radius.
• **Circumference:** The total distance around the circle's edge.

To effectively work with circles, one typically needs to use specific formulas. These allow measurements and properties like the circumference and area to be calculated accurately. This makes circle geometry not only practical but also crucial in both academic and real-world scenarios.

Diameter and Radius Relationship

Understanding the relationship between a circle's diameter and radius is key to solving many problems in circle geometry. Simply put, the diameter is always twice as long as the radius. This can be represented by the formula:

• If the diameter is known: \(d = 2r\)
• If the radius is known: \(r = \frac{d}{2}\)

For example, in a problem where the diameter is given as 12 inches, the radius can be calculated as:\[r = \frac{12}{2} = 6\text{ inches}\]Having a strong grasp of this relationship allows one to work interchangeably between the diameter and radius when applying formulas such as the circumference. This flexibility makes it easier to compute and solve problems effectively.

Mathematical Formula Manipulation

Manipulating mathematical formulas is a skill that simplifies complex problems. For circle-related calculations, the formula for circumference is essential: \[C = 2 \pi r\]When the diameter is provided instead of the radius, it's important to manipulate the formula to accommodate this change. By substituting the expression for the radius in terms of the diameter \(r = \frac{d}{2}\), we adjust the formula as follows:\[C = 2 \pi \left(\frac{d}{2}\right) = \pi d\]This operation replaces the radius in the standard circumference formula with the diameter, allowing direct calculations without needing a separate step to find the radius. This manipulation captures the same result faster and can be particularly helpful in complex problem-solving situations.