Chapter 9, Problem 43Q

To determine

The volume of ice in Antarctica and the amount by which the water level of the world’s oceans would rise, if the ice in Antarctica were to melt completely. Also, give the portions of the Earth that will get covered by water completely. It is given that the area of Antarctica is 13 million square km and the thickness of ice cap varies from 300 m at the coast to 1800 m in the interior.

Answer to Problem 43Q

Solution:

The volume of ice in Antarctica is $\text{13}{\text{.65×10}}^{\text{6}}{\text{ km}}^3$. Places such as coastal areas, islands, and other low-lying areas below 36 meters will submerge in water, if Antarctic ice were to melt completely.

Explanation of Solution

Given data:

The area of Antarctica is 13 million square kilometer or $13\times {10}^6{\text{ km}}^{\text{2}}$. The thickness (width) of the ice is 300 m near the coast and 1800 m in the interior. The ice and water have roughly the same density, which is ${\text{10}}^{\text{3}}{\text{ kg/m}}^{\text{3}}$.

Formula used:

The expression for volume is written as,

$V=Aw$

Here, V is the volume, A is the area and w is the width.

Density of a body is calculated as,

$D=\frac{M}{V}$

Here, M is mass of the body in kg and V is volume of the body in ${\text{kg}}^{\text{3}}$.

The expression for calculating area of a sphere is written as,

$A=4\pi r^2$

Here, r is the radius of the sphere.

Explanation:

The average thickness or width of the polar ice caps can be calculated by taking the average of the thickness given for the coastal and interior regions.

$\begin{array}{c}\text{Average thickness}=\frac{\text{300 m}+\text{1800 m}}{\text{2}}\\ =\text{1050 m}\end{array}$

Change the unit of width from m to km as,

$\begin{array}{c}\text{1000 m}=1\text{ km}\\ \text{1050 m}=\frac{\text{1050}}{\text{1000}}\\ \text{=1}\text{.05}\text{km}\end{array}$

Write the formula for volume of iceberg as,

$V=Aw$

Substitute 1.05 km for w and $13\times {10}^6{\text{km}}^{\text{2}}$ for A.

$\begin{array}{c}V=\left(13\times {10}^6{\text{km}}^{\text{2}}\right)\left(1.05\text{km}\right)\\ =13.65\times {10}^6{\text{ km}}^{\text{3}}\left(\frac{{10}^9{\text{ m}}^3}{1\text{ km}}\right)\\ =13.65\times {10}^{15}{\text{ m}}^3\end{array}$

Write the expression for density of water and rearrange for M.

$\begin{array}{c}D=\frac{M}{V}\\ M=DV\end{array}$

Substitute ${\text{10}}^{\text{3}}{\text{ kg/m}}^{\text{3}}$ for D and $13.65\times {10}^{15}{\text{m}}^{\text{3}}$ for V.

$\begin{array}{c}M=\left({10}^3{\text{ kg/m}}^{\text{3}}\right)\left(13.65\times {10}^{15}{\text{m}}^{\text{3}}\right)\\ =13.65\times {10}^{18}\text{kg}\end{array}$

Only 75% of Earth’s surface is covered in water. The radius of Earth is 6378 km. Therefore, the area of the Earth that will be covered will water can be calculated as:

$A=4\pi r^2$

Substitute 6378 km for r.

$\begin{array}{c}A=\frac{75}{100}\left(4\pi {\left(6378\text{ km}\right)}^2\right)\\ =3.83\times {10}^8{\text{ km}}^{\text{2}}\end{array}$

In case the ice melts, the rise in the water level on earth will be calculated using formula (1). Rearrange the variables in formula (1) as shown below:

$d=\frac{V}{A}$

Substitute $3.83\times {10}^8{\text{ km}}^{\text{2}}$ for A and $13.65\times {10}^6{\text{ km}}^{\text{2}}$ for V.

$\begin{array}{c}d=\frac{13.65\times {10}^6{\text{ km}}^{\text{2}}}{3.83\times {10}^8{\text{ km}}^{\text{2}}}\\ =0.036\text{ km}\left(\frac{1000\text{ m}}{1\text{ km}}\right)\\ =36\text{ m}\end{array}$

Conclusion:

Hence, the volume of ice in Antarctica is $\text{13}{\text{.65×10}}^{\text{6}}{\text{ km}}^{\text{3}}$. In case the ice of Antarctica melts, the depth of the oceans will increase by 36 m. Thus, the places below this level will submerge under water. The low-lying areas and the coastal regions and the islands will probably submerge under water in such a situation.