Problem 28 - Intermediate Algebra

Question

Find the circumference of each circle to the nearest hundredth. See Example 3. (Answers may vary slightly depending on which approximation of is used.) A circle with diameter \(6 \frac{1}{4} \mathrm{m}\)

Step-by-Step Solution

Answer

The circumference of the circle is approximately 19.63 meters.

Step 1: Convert Mixed Number to Improper Fraction

The diameter is given as a mixed number, \(6 \frac{1}{4}\). Convert it to an improper fraction by multiplying the integer part by the denominator of the fractional part and adding the numerator. Thus, \(6 \times 4 + 1 = 25\), so the diameter as an improper fraction is \(\frac{25}{4}\) meters.

Step 2: Recall the Formula for Circumference

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

Step 3: Plug Diameter into Circumference Formula

Substitute \(\frac{25}{4}\) for the diameter \(d\) in the formula \(C = \pi \times d\), giving us \(C = \pi \times \frac{25}{4}\).

Step 4: Approximate Using Known Value of \(\pi\)

We know that \(\pi \approx 3.14\) is a commonly used approximation. So we calculate \(C = 3.14 \times \frac{25}{4}\).

Step 5: Perform the Multiplication

First, multiply \(3.14\) by \(25\) to get \(78.5\). Then divide this product by \(4\) to find \(C = \frac{78.5}{4} = 19.625\).

Step 6: Round to the Nearest Hundredth

The result from the calculation is \(19.625\). To round to the nearest hundredth, we look at the thousandths place, which is \(5\). Since it is 5 or more, round up the hundredths place, giving us \(19.63\).

Key Concepts

Circle

A circle is a perfectly round shape where all points are equidistant from the center. A helpful way to think about a circle is that it is the set of all points in a plane that are the same distance from a given point, the center. This distance is known as the radius.

Diameter

The diameter of a circle is a straight line that passes from one side of the circle to the other, passing through the center point. It is the longest distance across the circle. The diameter is exactly twice the length of the radius or, stated mathematically, if the radius is denoted as \(r\), then the diameter \(d = 2r\). This makes the diameter critical when calculating the circumference, as the circumference \(C = \pi \times d\).

Improper Fractions

An improper fraction is a way to represent fractions where the numerator (the top number) is greater than or equal to the denominator (the bottom number). For instance, \(\frac{25}{4}\) is an improper fraction. Improper fractions are useful in mathematical calculations because they can make operations like multiplication and division simpler. If you start with a mixed number, such as \(6 \frac{1}{4}\), you can convert it to an improper fraction by multiplying the whole number by the fraction's denominator, adding the numerator, and placing the result over the original denominator. In \(6 \frac{1}{4}\), you compute \(6 \times 4 + 1 = 25\), so the improper fraction is \(\frac{25}{4}\).

Rounding Numbers

Rounding numbers is a way to simplify them, making them easier to work with while still being reasonably accurate. To round a number like 19.625 to the nearest hundredth, you follow a simple rule: look at the digit immediately after the place to which you are rounding—the thousandths place in this case, which is 5. If this digit is 5 or more, you'll round the hundredths place up by one. Here, the hundredths digit is 2, and since the thousandths digit is 5, you round it up, resulting in 19.63. This is helpful in contexts where precision is necessary, but overly precise numbers aren't practical.