Problem 63 - Prealgebra

Question

he radius of the earth is approximately \(3,900\) miles. Find the circumference of the earth at the equator. (The equator is a circle around the earth that divides the earth into two equal halves.)

Step-by-Step Solution

Answer

The circumference of the Earth at the equator is approximately 24,502.4 miles.

Step 1: Understanding the Problem

We need to find the circumference of the Earth at the equator given the radius of the Earth is approximately 3,900 miles. By definition, the equator is a circular line around the Earth, and we can use the formula for the circumference of a circle to find our answer.

Step 2: Applying the Formula for Circumference

The formula for the circumference of a circle is: \[ C = 2 \pi r \]where \( C \) is the circumference and \( r \) is the radius of the circle. In this case, \( r = 3,900 \) miles.

Step 3: Calculating the Circumference

Substitute the radius, \( r = 3,900 \) miles, into the formula:\[ C = 2 \pi \times 3,900 \].Perform the calculation to find the circumference.

Step 4: Performing the Multiplication

Multiply:\[ C = 7,800 \pi \] miles. To get a numerical value, calculate \( 7,800 \times \pi \) using \( \pi \approx 3.14159 \).

Step 5: Getting the Numerical Result

Calculate \( 7,800 \times 3.14159 \): \[ C \approx 24,502.4 \] miles.

Step 6: Conclusion

The circumference of the Earth at the equator is approximately 24,502.4 miles.

Key Concepts

Radius

The concept of radius is a fundamental aspect of geometry, especially when dealing with circles. The radius of a circle is the distance from the center point of the circle to any point on its edge. This measurement is crucial because it allows us to calculate other properties of the circle, such as the circumference and area.

• The radius is always the same length no matter where you measure from the center.
• In formulas, the radius is usually represented by the letter r.
• For large scale applications like the Earth's measurement, units such as miles or kilometers are commonly used.

In the given exercise, the Earth's radius is approximately 3,900 miles. This means that from the center of the Earth to its equatorial edge, it spans 3,900 miles, providing an essential component to calculate its circumference.

Pi Approximation

Understanding c0 (Pi) is key to working with circles. Pi (c0) is a mathematical constant representing the ratio of a circle's circumference to its diameter. As Pi is an irrational number, it cannot be accurately expressed as a simple fraction.

• The most common approximation of Pi is c0 248 3.14159.
• This constant remains the same, regardless of circle size.
• When using c0 in practical calculations, it's approximated to a few decimal places, often as 3.14.

In our exercise, Pi is used to convert the product of 2 times the radius into the circumference of the Earth. While c0 extends infinitely beyond the decimal, most calculations use the approximation 3.14159.

Circle Formula

The circle formula for circumference is invaluable in geometric calculations. The equation is expressed as \(C = 2 c0 r\),where \(C\) represents the circumference, \(r\) is the radius, and c0 is Pi. Here's how it works:

• This formula helps compute the length of how far it is around the circle.
• It's derived from multiplying the diameter (\(2r\)) by Pi (c0).
• In practical contexts, like measuring the Earth, it provides precise measurements needed for maps and navigational tools.

In the exercise, the formula reveals that the circle’s circumference, when given the Earth's radius of 3,900 miles, leads to the computation \(C = 7,800 c0\) miles before inserting Pi’s numerical value.

Earth's Equator

The Earth's equator is one of the most significant circles that exist on our planet. It is an imaginary line that divides the Earth into the Northern and Southern Hemispheres. Here are some key things to note about the equator:

• The equator is at equal distance from both Earth’s poles.
• Being approximately 24,902 miles in length, it serves as a baseline for geographical coordinates.
• It plays a critical role in climate systems and daylight distribution.

In this exercise, knowing the circumference of the Earth's equator helps in understanding Earth’s overall size. The numeral solution calculated was approximately 24,502.4 miles, which is a slight variation due to the usage of Pi's approximation, but it emphasizes the immense scale of this geographic feature.