Chapter 1, Problem 1P

To determine

To find:

a) Earth’s circumference in kilometers.

b) Earth’s surface area in square kilometers.

c) Earth’s volume in cubic kilometers.

Answer to Problem 1P

Solution:

a) Earth’s circumference is $4.00\times {10}^4\mathrm{ }\mathrm{k}\mathrm{m}$.

b) Earth’s surface area is $5.10\times {10}^8{\mathrm{k}\mathrm{m}}^2$.

c) Earth’s volume is $1.08\times {10}^{12}\mathrm{k}{\mathrm{m}}^3$.

Explanation of Solution

Given:

Radius of the earth: $6.37\times {10}^{6}m$.

Calculations:

a) The circumference of sphere is $2\mathrm{\pi }\mathrm{R}$.

So, the circumference of Earth  $=2\mathrm{\pi }\mathrm{R}=2\times \mathrm{\pi }\times 6.37\times {10}^6$.

$\mathrm{C}\mathrm{i}\mathrm{r}\mathrm{c}\mathrm{u}\mathrm{m}\mathrm{f}\mathrm{e}\mathrm{r}\mathrm{e}\mathrm{n}\mathrm{c}\mathrm{e}\mathrm{ }\mathrm{o}\mathrm{f}\mathrm{ }\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{ }\mathrm{e}\mathrm{a}\mathrm{r}\mathrm{t}\mathrm{h}=4\times {10}^7\mathrm{m}$.

But, 1 km = 1000 m.

Therefore, the circumference of the Earth is $4.00\times {10}^4\mathrm{ }\mathrm{k}\mathrm{m}$.

b) The Surface area of sphere = $4\mathrm{\pi }{\mathrm{R}}^2$.

So, the surface area of Earth = $4\mathrm{\pi }{\mathrm{R}}^2$

= $4\times \mathrm{\pi }\times {\left(6.37\times {10}^6\right)}^2$

= $5.10\times {10}^{14}\mathrm{m}$

But, ${\left(1\mathrm{ }\mathrm{k}\mathrm{m}\right)}^2={10}^6{\mathrm{m}}^2$

Therefore, the surface area of the Earth is $5.10\times {10}^8{\mathrm{k}\mathrm{m}}^2$.

c) Volume of sphere = $\frac{\mathrm{ }4\mathrm{ }\mathrm{\pi }{R}^3}{3}$

So, the volume of Earth $\left(V\right)=\frac{\mathrm{ }4\mathrm{ }\mathrm{\pi }{\mathrm{R}}^3}{3}$

$V=\frac{4}{3}\mathrm{\pi }{\left(6.37\times {10}^6\right)}^3$

$V=1.08\times {10}^{21}{\mathrm{m}}^3$

But, ${\left(1\mathrm{ }\mathrm{k}\mathrm{m}\right)}^3={10}^9{\mathrm{m}}^3$

Therefore, the volume of the Earth is $1.08\times {10}^{12}\mathrm{k}{\mathrm{m}}^3$.

Conclusion:

Using the formula for the circumference of the sphere, we can find the earth’s circumference in meters and convert it in kilometers. So, the answer in km is $\mathrm{ }4.00\times {10}^4\mathrm{ }\mathrm{k}\mathrm{m}$.

Using the formula for the surface area of the sphere, we can find the Earth’s surface area and convert this from square meters to square kilometers. So the surface area of the Earth is $5.10\times {10}^8{\mathrm{k}\mathrm{m}}^2$.

And finally, using the formula for the volume of the sphere, we can find the volume of the Earth in cube meters and convert it into cube kilometers. So the volume of the earth in cube kilometer is $\mathrm{ }1.08\times {10}^{12}\mathrm{k}{\mathrm{m}}^3$.