Q1P - Fundamentals Of Physics

Question

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

Step-by-Step Solution

Answer

Answer

 

1. Earth’s circumference is$4.00\times {10}^{4}km$.
2. Earth’s surface area is$5.10\times {10}^{8}k{m}^2$ .
3. Earth’s volume is $1.08\times {10}^{12}k{m}^3$.

 

Step 1: Given data

The radius of the Earth, $=6.37\times {10}^{6}m$

Step 2: Understanding the system of units

A system of units is a set of units in which units are represented in terms of some fundamental units. In SI system of units, there are seven fundamental units.

 

The expression for the circumference is given as follows: 

 

   $C=2\pi R$                                                                                     … (i)

 

The expression for the surface area of sphere is given as follows: 

 

          $A=4\pi R^2$                                                                            … (ii)

 

The expression for the volume of sphere is given as follows: 

 

             $V=\frac{4\pi R^3}{3}$                                                                       … (iii)

Step 3: (a) Determination of Circumference of the Earth

Convert the radius of the Earth from meter to kilometer. 

 $$\begin{gathered} R=6.37\times {10}^{6}m \\ =6.37\times {10}^{6}m\times \frac{1km}{1000m} \\ =6.37\times {10}^{3}km \end{gathered}$$

 

Substitute the value of R in equation (i). 

 $$\begin{gathered} C=2\pi R \\ =2\times 3.14\times 6.37\times {10}^{6}m \\ =4.00\times {10}^{4}km \end{gathered}$$

 

Thus, the circumference of the Earth is $4.00\times {10}^{4 }km$.

Step 4: (b) Determination of the surface area of a sphere

$$\begin{gathered} \text{Using equation (ii), the surface area of Earth is calculated as: } \\ A=4{\mathrm{\pi R}}^2 \\ =4\times 3.14\times (6.37\times {10}^3) \\ =5.10\times {10}^{8 }{\mathrm{km}}^2 \\ \text{ Thus, the surface area of the Earth is }5.10\times {10}^{8 }{\mathrm{km}}^2\text{ .} \end{gathered}$$

Step 5: (c) Determination of the volume of the Earth

$$\begin{gathered} \text{Using equation (iii), the volume of the Earth is calculated as: } \\ V=\frac{4\pi R^3}{3} \\ =\frac{4\times 3.14\times (6.37\times {10}^{3}km)}{3} \\ =1.08\times {10}^{12}km \\ \text{ Thus, the volume of the Earth is }1.08\times {10}^{12}km\text{ .} \end{gathered}$$