Problem 64 - Prealgebra

Question

The radius of the moon is approximately \(1,100\) miles. Find the circumference of the moon around its equator.

Step-by-Step Solution

Answer

The circumference of the moon is approximately 6,911.676 miles.

Step 1: Recall the Formula for Circumference

The formula for the circumference of a circle is given by \( C = 2 \pi r \), where \( C \) is the circumference and \( r \) is the radius.

Step 2: Identify the Radius

The given radius of the moon is \( 1,100 \) miles. This is the value we'll use for \( r \) in our formula.

Step 3: Plug in the Value into the Formula

Using the formula \( C = 2 \pi r \), substitute \( r = 1,100 \). This gives us \( C = 2 \pi \times 1,100 \).

Step 4: Calculate the Expression

Calculate the expression \( 2 \pi \times 1,100 \). Using the approximation \( \pi \approx 3.14159 \), calculate: \( C \approx 2 \times 3.14159 \times 1,100 \).

Step 5: Simplify to Find the Circumference

Perform the multiplication: \( 2 \times 3.14159 \times 1,100 \approx 6,911.676 \). Therefore, the circumference of the moon is approximately \( 6,911.676 \) miles.

Key Concepts

Radius of a Circle

The radius of a circle is an essential part of understanding circles. Simply put, the radius is the distance from the center of the circle to any point on the circumferential edge. It's half the length of the diameter, which spans from one edge of the circle through the center to the opposite edge.
If you're dealing with a circle and know its diameter, you can easily find the radius by dividing the diameter by two. This makes calculations involving circles much simpler!

• For instance, if a circle has a diameter of 10 miles, its radius is 5 miles.
• It's also important to note that in any circle, all radii have the same length.

In the context of our exercise, the radius of the moon used is 1,100 miles. This measurement is crucial as it allows us to compute other properties of the moon's circle, such as its circumference.

Circumference Formula

The circumference of a circle might sound complex, but it's actually quite straightforward with the right formula! The circumference, which is the distance around the circle, can easily be calculated if you know the radius.
Here's the formula:

• \[ C = 2 \pi r \]

Where \( C \) represents the circumference, \( \pi \) (Pi) is a constant approximately equal to 3.14159, and \( r \) stands for the radius.
For any circle, this formula allows you to find the length of the boundary line enclosing the circle. By plugging the radius into the formula, you can effortlessly obtain the circumference!
In our specific problem with the moon, using this formula lets us find how big around the moon is at its equator, using the radius we discussed earlier.

Approximation of Pi

Pi, commonly represented by the symbol \( \pi \), is a very special and mysterious number in mathematics. It represents the ratio of a circle's circumference to its diameter and is approximately 3.14159. Its approximation is often used in calculations for ease, even though Pi is an irrational number which means it has infinite decimal places and cannot be expressed exactly as a fraction.
Using approximate values of Pi is practical and sufficient for most calculations. In practical terms:

• We may use \( \pi \approx 3.14 \) for simpler estimations.
• In more precise scientific calculations, a longer decimal like \( \pi\approx 3.14159 \) is typically used.

In our exercise on the moon’s circumference, we use the approximation \( \pi \approx 3.14159 \) to find a precise circumference. Understanding Pi helps us realize why circles, including planets and moons, have their mysterious proportions!