Problem 47 - Basic Technical Mathematics with Calculus

Question

The radius of the Earth's equator is 3960 mi. What is the circumference?

Step-by-Step Solution

Answer

The circumference of the Earth's equator is approximately 24,881.6 miles.

Step 1: Understand the Formula

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

Step 2: Identify the Parameters

For this exercise, the radius \(r\) of the Earth's equator is given as 3960 miles. We will use this value in our formula.

Step 3: Plug in the Radius Value

Substitute the radius \(r = 3960\) mi into the circumference formula: \(C = 2\pi \times 3960\).

Step 4: Calculate the Circumference

Evaluate the expression \(C = 2\pi \times 3960\), which equals \(C \approx 2 \times 3.1416 \times 3960\). Using a calculator, \(C \approx 24881.6\) miles.

Key Concepts

Radius

The term "radius" refers to the distance from the center of a circle to any point on its edge. In simple terms, it's like drawing a straight line from the middle of a circle to the edge. This distance is crucial in circle-related calculations.
In mathematical problems, especially those involving circles, the radius is key to finding other properties like the diameter and circumference. For example:

• The diameter is twice the radius: \(d = 2r\).
• The circumference can be found if you have the radius: \(C = 2\pi r\).

The radius is vital because it serves as a fundamental building block in the formulas used to discover other attributes of circles. When given the radius of the Earth's equator, like in our example, we can use it to compute the distance around the equator, known as the circumference.

Earth's Equator

The Earth's equator is an imaginary line that circles the Earth horizontally. It is equidistant from the North and South Poles and divides the Earth into the Northern and Southern Hemispheres.
Because of its unique position, the equator is significant in geographical and physical studies. It has the longest circumference of any latitude line since it passes through the widest part of the Earth.
When calculating the size of the Earth's equator, the radius of the equator is needed, which is quite vast at 3960 miles. Knowing the equatorial radius allows us to determine important measures, like the circumference, using mathematical formulas.

Mathematical Formula

In mathematics, formulas are like recipes—they give you the steps and ingredients needed to solve a problem. For calculating the circumference of a circle, we use the formula:
\[C = 2\pi r\]Here, \(C\) represents the circumference, \(r\) is the radius, and \(\pi\) is a constant approximately equal to 3.1416.
Formulas are essential because they provide a standard method for solving similar problems and understanding relationships between different quantities. In this example, using the equatorial radius of the Earth, we applied the circumference formula to secure a consistent and repeatable result.
Thus, formulas not only solve problems but also reveal patterns and connections in nature and science.

Circle

A circle is a perfectly round shape where every point on the edge is the same distance from its center. This constant distance is what we refer to as the radius.
Circles are basic shapes studied in geometry, and their symmetry makes them fascinating subjects in both math and art. Characteristics of circles:

• The diameter is the longest straight line you can draw across the circle.
• The area is the space enclosed by the circle, calculated by \(A = \pi r^2\).

Circles arise in various real-world scenarios, including wheels, clocks, and even the orbits of planets. To describe a circle's size or boundary, we often resort to the circumference. The study and understanding of circles can illuminate important mathematical concepts and have practical applications in engineering, architecture, and various sciences.