Problem 49 - Elementary Algebra

Question

Set up an algebraic equation and then solve. The diameter of a circle measures 5 inches. Determine the circumference to the nearest tenth.

Step-by-Step Solution

Answer

The circumference is approximately 15.7 inches.

Step 1: Identify Relevant Information

We are given the diameter of a circle, which measures 5 inches. We need to use this to find the circumference of the circle.

Step 2: Recall the Formula for Circumference

The formula for the circumference of a circle, when the diameter is known, is given by \( C = \pi \times d \), where \( C \) is the circumference and \( d \) is the diameter.

Step 3: Substitute the Diameter into the Formula

Substitute the given diameter into the formula: \( C = \pi \times 5 \).

Step 4: Calculate the Circumference

Using an approximation for \( \pi \), such as 3.14, \( C = 3.14 \times 5 = 15.7 \).

Step 5: Round to the Nearest Tenth

The calculated circumference, 15.7, is already to the nearest tenth. Thus, the circumference of the circle is 15.7 inches.

Key Concepts

Understanding Diameter

The diameter of a circle is a straight line passing through the center of the circle, connecting two points on its boundary. It is one of the most essential parameters when discussing circles.
Understanding the diameter is crucial because it relates directly to the circumference and area of the circle.
Here’s why the diameter is important:

• The diameter is precisely twice the length of the radius (the distance from the center of the circle to any point on its edge).
• It can be expressed as: \( d = 2r \), where \( r \) is the radius.
• Knowing the diameter allows you to easily calculate the circle's circumference and area.

In our problem, the diameter of the circle is given as 5 inches. This information is key to determining the circumference through the proper use of formulas.

Formulating an Algebraic Equation

An algebraic equation is a mathematical statement that signifies the equality of two expressions. It usually involves variables and numbers.
In the context of finding the circumference of a circle, the algebraic equation comes from substituting the known values into a standard formula. For finding the circumference when the diameter is known, use the equation:

• \( C = \pi \times d \), where \( C \) stands for circumference and \( d \) represents the diameter.

This formulation allows us to solve directly for the circumference by inserting the known value of the diameter.
So, when our problem provides a diameter of 5 inches, the algebraic equation becomes \( C = \pi \times 5 \). This step sets the mathematical foundation to find the circumference.

The Concept of Pi

Pi (\( \pi \)) is a mathematical constant that represents the ratio of a circle's circumference to its diameter. This fascinating number is approximately 3.14159, though it continues infinitely without repeating.Here's what makes pi essential for circle calculations:

• Pi is used in many formulas involving circles, including circumference and area.
• In practical calculations, pi is often approximated as \( 3.14 \), or more precisely \( \frac{22}{7} \).

For the exercise at hand, pi is used to convert diameter measurements into circumference measurements using the formula \( C = \pi \times d \).
By approximating pi as 3.14, it simplifies our calculations without losing significant accuracy.
Understanding pi and how it interacts with the diameter and circumference is pivotal to solving geometric problems effectively.