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Chapter 6: Problem 59

If Earth is a sphere with a radius of 3884.3 miles, find the volume of Earth.

Short Answer

Expert verified
Question: Calculate the approximate volume of Earth, given that its radius is 3884.3 miles. Answer: The approximate volume of Earth is 2.4505 * 10^11 cubic miles.

Step by step solution

01

Identify the given values

The radius of Earth is given as 3884.3 miles.
02

Write down the formula for the volume of a sphere

The formula for the volume of a sphere is: V = \frac{4}{3} * \pi * r^3
03

Substitute the given radius into the formula

Replace r with the given radius of Earth, 3884.3 miles: V = \frac{4}{3} * \pi * (3884.3)^3
04

Calculate the volume of Earth

Perform the calculations in the formula: V = \frac{4}{3} * \pi * (3884.3)^3 鈮� 2.4505 * 10^11 cubic miles So, the volume of Earth is approximately 2.4505 * 10^11 cubic miles.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Radius
The radius is a fundamental aspect of a sphere. It is the distance from the center of the sphere to any point on its surface. In any sphere, the radius is the same from the center to any spot on the exterior, making it a defining property. For Earth, this measurement is quite substantial, at 3884.3 miles as given in the exercise.
Knowing the radius is crucial because it serves as a key component in calculating the sphere's volume and surface area. When dealing with spheres, starting with an accurate radius measurement ensures that subsequent calculations for properties like volume will be accurate.
The Concept of a Sphere
A sphere is a perfectly symmetrical three-dimensional shape, where every point on the surface is equidistant from the center. This uniformity makes spheres quite special in mathematics and physics. They have no edges or vertices, differentiating them from cubes or pyramids.
The symmetry of a sphere simplifies many calculations and is an ideal geometric shape for applications like planetary bodies, such as Earth. Understanding spheres helps us apply mathematical concepts to real-world objects effectively.
  • Spheres have consistent curvatures.
  • They are often used in modeling celestial bodies.
  • They are defined primarily by their radius.
Mathematical Formula Insights
The volume of a sphere is determined using an important mathematical formula: \[ V = \frac{4}{3} \pi r^3 \]This formula arises from integral calculus and is fundamental in geometry to understand how much space a sphere encompasses. Knowing this formula is vital because it simplifies a complex three-dimensional shape down to a straightforward calculation that can be used for various applications.
In this formula,
  • \(V\) represents the volume.
  • \(\pi\) is a constant approximately equal to 3.14159.
  • \(r\) is the radius of the sphere.
Understanding the formula helps in predicting how changes in radius can significantly impact the volume of the sphere.
Steps in Calculation
Performing calculations with the formula for the volume of a sphere involves a systematic approach. First, identify the given radius. Here, it's 3884.3 miles for Earth.
Next, insert this radius into the formula: \[ V = \frac{4}{3} \pi (3884.3)^3 \]Now, complete the calculations:
  • First, cube the radius: \( (3884.3)^3 \)
  • Next, multiply by \(\pi\) and then by \(\frac{4}{3}\).
These calculations result in the volume of Earth, approximately 2.4505 * 10^11 cubic miles. The sequential steps ensure accuracy and clarity, important in complex calculations like finding the volume of a planet.

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