Problem 92 - Prealgebra

Question

A circle has a diameter of \(14.11\) 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 44.3 inches.

Step 1: Understand the Circumference Formula

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

Step 2: Identify the Given Diameter

From the problem statement, we know the diameter of the circle is \( 14.11 \) inches.

Step 3: Substitute the Given Values

Insert the given values into the circumference formula. This gives us \( C = 3.14 \times 14.11 \).

Step 4: Calculate the Circumference

Perform the multiplication to find the circumference. Calculate \( 3.14 \times 14.11 \) to get approximately \( 44.2854 \).

Step 5: Round to the Nearest Tenth

Round \( 44.2854 \) to the nearest tenth. To do this, look at the hundredths place, which is 8. Since it is greater than 5, round up the tenths place from 2 to 3, giving us \( 44.3 \).

Key Concepts

Diameter

The diameter is a key measurement of a circle that helps us understand the size of the circle. It is the straight line passing from one side of the circle to the other through the center. Essentially, the diameter measures the widest part of the circle. The diameter is crucial in calculating the circumference of a circle, acting as one of the main variables in the formula.

• Diameter is twice the radius: If you know the radius (half of the diameter), you can simply double it to determine the diameter.
• The formula for circumference using diameter is: \( C = \, \pi \times d \)

For example, in our exercise, we know that the diameter of the circle is \( 14.11 \) inches. By using this value in the circumference formula, we can determine how long the distance around the circle is.

Pi Approximation

Pi is a mathematical constant representing the ratio of a circle's circumference to its diameter. It's an irrational number, meaning it cannot be exactly expressed as a simple fraction, and its decimal representation goes on forever without repeating. However, for practical calculations, we often use approximations of Pi.

• Common approximations include \( \pi \approx 3.14 \), \( \pi \approx \frac{22}{7} \), or even further decimal places like 3.14159 for more precision.
• Using \( \pi \approx 3.14 \) is standard for many simple calculations since it provides a balance between simplicity and accuracy.

In our exercise, we approximate Pi as 3.14, which is a typical substitute in everyday tasks and aids in understanding how changes in Pi's value might affect the final circumference result of a circle.

Rounding Numbers

Rounding numbers is essential in mathematics to simplify complex decimal numbers, making them easier to understand and use. When we round to the nearest tenth, we focus on the first number after the decimal point and decide whether to round it up or keep it as is.

• If the number in the hundredths place (second after the decimal) is 5 or greater, round up.
• If it's less than 5, keep the tenths place the same.

In our exercise, we calculated a circumference value of approximately \( 44.2854 \). Here, the tenths place is 2, and the hundredths place is 8. Since 8 is greater than 5, we round up the tenths place from 2 to 3, resulting in a more simplified and easily comprehensible 44.3 inches.