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Chapter 1: Problem 13

Earth is approximately a sphere of radius \(6.37 \times\) \(10^{6} \mathrm{~m}\). What are (a) its circumference in kilometers, (b) its surface area in square kilometers, and (c) its volume in cubic kilometers?

Short Answer

Expert verified
Circumference: \ 4.00 \times 10^4 \mathrm{~km}\, Surface Area: \ 5.12 \times 10^8 \mathrm{~km^2} \ , Volume: \ 1.08 \times 10^{12} \mathrm{~km^3}\

Step by step solution

01

Understand the problem

We need to find the circumference, surface area, and volume of a sphere given its radius. The radius of the Earth is provided as \(6.37 \times 10^6 \mathrm{~m}\).
02

Convert radius to kilometers

Convert the radius from meters to kilometers. Since 1 kilometer is 1000 meters, the radius in kilometers is \( \frac{6.37 \times 10^6}{1000} = 6.37 \times 10^3 \mathrm{~km}\).
03

Calculate the circumference

The formula for the circumference of a sphere is \(C = 2 \pi r\). Substitute \(r = 6.37 \times 10^3\) kilometers into the formula: \[ C = 2 \pi (6.37 \times 10^3) = 2 \times 3.14159 \times 6.37 \times 10^3 = 4.00 \times 10^4 \mathrm{~km} \]
04

Calculate the surface area

The formula for the surface area of a sphere is \(A = 4 \pi r^2\). Substitute \(r = 6.37 \times 10^3\) kilometers into the formula: \[ A = 4 \pi (6.37 \times 10^3)^2 = 4 \times 3.14159 \times (6.37 \times 10^3)^2 \] First calculate \( (6.37 \times 10^3)^2 = 4.06 \times 10^7\). Now substitute: \[ A = 4 \times 3.14159 \times 4.06 \times 10^7 = 5.12 \times 10^8 \mathrm{~km^2} \]
05

Calculate the volume

The formula for the volume of a sphere is \(V = \frac{4}{3} \pi r^3\). Substitute \(r = 6.37 \times 10^3\) kilometers into the formula: \[ V = \frac{4}{3} \pi (6.37 \times 10^3)^3 = \frac{4}{3} \times 3.14159 \times (6.37 \times 10^3)^3 \] First calculate \( (6.37 \times 10^3)^3 = 2.58 \times 10^{11}\). Now substitute: \[ V = \frac{4}{3} \times 3.14159 \times 2.58 \times 10^{11} = 1.08 \times 10^{12} \mathrm{~km^3} \]

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Key Concepts

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

circumference calculation
Let's start by understanding what the circumference of a sphere means. The circumference is the distance around the sphere along a great circle, which is a circle that slices through the center of the sphere. To calculate the Earth's circumference, we use the formula: \( C = 2 \pi r \).

Here, \( r \) represents the radius of the Earth. First, you need to convert the given radius from meters to kilometers because the later calculations will be in kilometers. Since \(1\) kilometer is \(1000\) meters:

\[ r = \frac{6.37 \times 10^6 \text{ meters}}{1000} = 6.37 \times 10^3 \text{ km} \]

Now, substitute this radius back into the formula:

\[ C = 2 \pi (6.37 \times 10^3) = 2 \times 3.14159 \times 6.37 \times 10^3 = 4.00 \times 10^4 \text{ km} \]

This tells us that the Earth's circumference is approximately \(4.00 \times 10^4\) kilometers.
surface area formula
Next, we'll look at the surface area of a sphere. The surface area is the total area that the surface of the sphere occupies, and it's calculated using the formula: \( A = 4 \pi r^2 \).

Again, we use the Earth's radius converted to kilometers: \(6.37 \times 10^3\) kilometers. Substitute the radius back into the formula:

\[ A = 4 \pi (6.37 \times 10^3)^2 \]

First, calculate \((6.37 \times 10^3)^2\):

\[ (6.37 \times 10^3)^2 = 4.06 \times 10^7 \]

Now, substitute this back into the surface area formula:

\[ A = 4 \times 3.14159 \times 4.06 \times 10^7 = 5.12 \times 10^8 \text{ km}^2 \]

This means that the Earth's surface area is approximately \(5.12 \times 10^8\) square kilometers.
volume computation
Finally, let's find out the volume of the Earth, which is the amount of space the Earth occupies. The volume of a sphere is calculated using the formula: \( V = \frac{4}{3} \pi r^3 \).

We'll use the radius in kilometers once again: \(6.37 \times 10^3\) kilometers. Place the radius into the formula:

\[ V = \frac{4}{3} \pi (6.37 \times 10^3)^3 \]

Calculate \((6.37 \times 10^3)^3\) first:

\[ (6.37 \times 10^3)^3 = 2.58 \times 10^{11} \]

Now substitute back into the volume formula:

\[ V = \frac{4}{3} \times 3.14159 \times 2.58 \times 10^{11} = 1.08 \times 10^{12} \text{ km}^3 \]

Therefore, the Earth's volume is approximately \(1.08 \times 10^{12}\) cubic kilometers.

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Most popular questions from this chapter

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