Problem 91 - Prealgebra

Question

A circle has a diameter of \(12.04\) 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 37.8 inches.

Step 1: Understanding the Circumference Formula

The formula to calculate the circumference of a circle is given by \( C = \pi \times d \), where \( C \) is the circumference and \( d \) is the diameter. Here, \( \pi \approx 3.14 \).

Step 2: Substitute the Given Diameter

In this exercise, the diameter \( d \) is given as 12.04 inches. Substitute this value into the circumference formula: \( C = 3.14 \times 12.04 \).

Step 3: Perform the Multiplication

Multiply the numbers: \( 3.14 \times 12.04 \). First, calculate \( 3.14 \times 12.04 = 37.8056 \).

Step 4: Round to the Nearest Tenth

Round the result from the previous step to the nearest tenth. The number 37.8056 rounds to 37.8 when considering the digits after the decimal point.

Key Concepts

Understanding Diameter

The diameter of a circle is an essential concept in geometry. It refers to the longest distance from one edge of a circle to the other, passing through the center point. Think of it as a straight line that cuts the circle in half perfectly.
This line is crucial in calculations involving circles, especially when finding the circumference. Given how the diameter relates to the circle's size, it helps determine many other properties of the circle.
In our problem, the diameter is given as 12.04 inches. Knowing this allows us to compute the circle’s circumference by using the formula involving the diameter and pi. Anytime you have the diameter, calculating the circumference is just a step away using the right formula.

Pi Approximation

Pi (π) is a mathematical constant that represents the ratio of a circle's circumference to its diameter. This constant is approximately 3.14159, but it's often rounded to simpler forms for practical calculations.
In many typical problems, especially where precision isn't crucial, we use the approximation of \(pi \approx 3.14\). This makes calculations much easier and faster, mainly when using pen and paper or performing mental math.
Using pi in this way is handy because it allows us to estimate the circumference based on the diameter, as seen in our exercise. With \(pi = 3.14\), multiplying it by the diameter gives us a good approximation of the circle's circumference.

Rounding Decimals

Rounding decimals is a way of simplifying a number while maintaining its overall value. It's especially useful when you need less precision or when it's more convenient to work with fewer decimal places.
You round a number by looking at the digit immediately after the place you're rounding to. If this digit is five or more, you round up the preceding digit. If it's less than five, you keep the preceding digit the same.

• For instance, if rounding to the nearest tenth, you focus on the digits immediately after the first decimal place.
• In our exercise, 37.8056 becomes 37.8 when rounded to the nearest tenth. The '0' tells us to leave the '8' unchanged.

This process helps clear confusion, especially in making quick and accurate measurements. In practical applications, such as our example with circle circumference, rounding provides a result that's easy to interpret and apply.