Problem 90 - Prealgebra

Question

A circle has a diameter of \(14.23\) inches. Using \(\pi \approx 3.14\), find the circumference of the circle, correct to the nearest tenth of an inch.

Step-by-Step Solution

Answer

The circumference is approximately 44.7 inches.

Step 1: Identify the Given Values

We are given that the diameter of the circle is 14.23 inches and we have to use \(\pi \approx 3.14\).

Step 2: Recall the Formula for Circumference

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

Step 3: Calculate the Circumference

Substitute the known values into the formula: \(C = 3.14 \times 14.23\). Perform the multiplication to find \(C\).

Step 4: Perform the Multiplication

Calculate \(3.14 \times 14.23 = 44.6822\).

Step 5: Round to the Nearest Tenth

Round the result to the nearest tenth: \(44.7\) inches.

Key Concepts

Understanding Diameter

To grasp the concept of circle circumference, you need to first understand what diameter is. The diameter of a circle is the distance across the circle, passing through its center. It connects one side of the circle to the other directly through the middle. This means that the diameter is twice the length of the radius, which is the distance from the center to any point on the edge of the circle. In mathematical terms, if the radius is denoted as \( r \), then the diameter \( d \) can be calculated as: \[ d = 2r \] Knowing the diameter is essential for calculating the circumference of a circle, as it directly plugs into the circumference formula. Always make sure you have the correct diameter before proceeding to calculate other properties of a circle. For our exercise, the diameter is given as 14.23 inches. With this key measurement, we can accurately determine other attributes, such as the circumference, when \( \pi \) is given.

Approximating Pi

The number pi (\( \pi \)) is a special mathematical constant that represents the ratio of any circle's circumference to its diameter. It is an irrational number, meaning it has an infinite number of non-repeating decimals. For simplicity in basic calculations, \( \pi \) is often approximated. Common approximations include:

• 3.14
• 22/7

Using an approximation of \( \pi \) allows us to perform calculations without dealing with an endless decimal. In this exercise, \( \pi \approx 3.14 \) is used to make the arithmetic simpler and still maintain accuracy within reasonable bounds. This approximation is widely accepted and commonly used in school exercises and various practical settings.

Rounding Decimals

Rounding decimals is an essential skill in simplifying numerical results, especially in estimations and real-word applications. When given a long decimal result, rounding helps to make it more concise and easier to interpret. Here’s how you can round a number to the nearest tenth: 1. Identify the digit at the tenths place, which is the first digit to the right of the decimal point. 2. Check the digit in the hundredths place, which is the second digit to the right of the decimal point. 3. If the hundredths digit is 5 or greater, increase the tenths digit by 1. If it is less than 5, leave the tenths digit as it is. For our exercise, the circumference was calculated to be 44.6822 inches. By reviewing the hundredths digit (8), which is greater than 5, we increase the tenths digit, rounding the result to 44.7 inches. This process helps simplify the result, especially in settings where excessive precision is not necessary.