Chapter 7.3, Problem 66PPE

(a).

To determine

To find: The approximate surface area of the Earth as scientific notation.

Answer to Problem 66PPE

The approximate surface area of the Earth is $5.15\times {10}^{14}{\text{m}}^{\text{2}}$ .

Explanation of Solution

Given information: The radius of the Earth is $6.4\times {10}^6\text{m}$ .

Calculation:

The formula for the surface area of a sphere with radius r is given by,

  $S.A.=4\pi r^2$

Substitute $6.4\times {10}^6\text{m}$ for r in the above formula.

  $S.A.=4\pi {\left(6.4\times {10}^6\text{m}\right)}^2$

The property of raising a product to a power for any positive number $n$ is:

  ${\left(ab\right)}^n=a^n{b}^n$

To determine the surface area, apply the above property.

  $\begin{array}{c}S.A.=4\pi {\left(6.4\right)}^2\times {\left({10}^6\right)}^2{\text{m}}^2\\ =4\left(3.14\right)\left(40.96\right)\times {\left({10}^6\right)}^2{\text{m}}^2\\ =514.4576\times {\left({10}^6\right)}^2{\text{m}}^2\end{array}$

Apply a power of a power property of exponents to further simplify.

  $\begin{array}{c}S.A.=514.4576\times {\left(10\right)}^{6\times 2}{\text{m}}^2\left[\therefore {\left(a^m\right)}^n={\left(a\right)}^{m\times n}\right]\\ =514.4576\times {10}^{12}{\text{m}}^{\text{2}}\\ \approx 5.15\times {10}^{14}{\text{m}}^{\text{2}}\end{array}$

Therefore, the approximate surface area of the Earth in scientific notation is $5.15\times {10}^{14}{\text{m}}^{\text{2}}$ .

(b).

To determine

To find: The area of Earth’s surface covered by ocean water in square meters.

Answer to Problem 66PPE

The area of Earth’s surface covered by ocean water in square meters is $3.605\times {10}^{14}$ .

Explanation of Solution

Given information: The area of Earth surface covered by ocean in per cent is 70%.

Calculation:

As calculated in part (a), the approximate surface area of the Earth is $5.15\times {10}^{14}{\text{m}}^{\text{2}}$ .

The area of Ocean water can be calculated as:

  $\begin{array}{c}\text{Area}\text{of}\text{}\text{Ocean}\text{water}=70\%\times \text{area}\text{of}\text{Earth's}\text{surface}\\ \text{=}\frac{70}{100}\times \text{area}\text{of}\text{Earth's}\text{surface}\\ =0.7\times \text{area}\text{of}\text{Earth's}\text{surface}\end{array}$

Substitute $5.15\times {10}^{14}{\text{m}}^{\text{2}}$ for Earth’s surface area in the above expression.

  $\begin{array}{c}\text{Area}\text{of}\text{}\text{Ocean}\text{water}=0.7\times 5.15\times {10}^{14}{\text{m}}^{\text{2}}\\ =3.605\times {10}^{14}{\text{m}}^{\text{2}}\end{array}$

Therefore, the area of Earth’s surface covered by ocean water in square meters is $3.605\times {10}^{14}$ .

(c).

To determine

To find: The estimated volume of ocean water in Earth’s surface.

Answer to Problem 66PPE

The estimated volume of ocean water in Earth’s surface is $1.36\times {10}^{18}{\text{m}}^3$ .

Explanation of Solution

Given information: The average depth of ocean is 3790 m.

Calculation:

As calculated in part (b), the area of Earth’s surface covered by ocean water in square meters is $3.605\times {10}^{14}{\text{m}}^{\text{2}}$ .

The formula to determine volume of ocean water in Earth’s surface is:

  $V=S.A\times \text{depth}$

Substitute $3.605\times {10}^{14}{\text{m}}^{\text{2}}$ for surface area and 3790 m for depth in the above formula.

  $\begin{array}{c}V=3.605\times {10}^{14}{\text{m}}^{\text{2}}\times 3790\text{m}\\ \text{=13662}\text{.95}\times {10}^{14}{\text{m}}^3\\ \approx 1.36\times {10}^{18}{\text{m}}^3\end{array}$

Therefore, the estimated volume of ocean water in Earth’s surface is $1.36\times {10}^{18}{\text{m}}^3$ .