Problem 79 - Algebra and Trigonometry

Question

Volume of the Oceans The average ocean depth is \(3.7 \times 10^{3} \mathrm{m},\) and the area of the oceans is \(3.6 \times 10^{14} \mathrm{m}^{2}\) . What is the total volume of the ocean in liters? (One cubic meter contains 1000 liters.)

Step-by-Step Solution

Answer

The total ocean volume is approximately \(1.332 \times 10^{21}\) liters.

Step 1: Understand the Formula for Volume

The formula to calculate the volume of a body of water is given by \( \text{Volume} = \text{Area} \times \text{Depth} \). We need the average depth and surface area to find the volume of the oceans.

Step 2: Substitute Given Values into the Formula

We know the average ocean depth is \( 3.7 \times 10^3 \text{ m} \) and the area of the ocean is \( 3.6 \times 10^{14} \text{ m}^2 \). Substituting these values into the formula gives: \[ \text{Volume} = (3.6 \times 10^{14}) \times (3.7 \times 10^3). \]

Step 3: Calculate the Volume in Cubic Meters

To find the volume in cubic meters, multiply the coefficients and add the exponents: \[3.6 \times 3.7 = 13.32 \] and \(10^{14} \times 10^3 = 10^{17} \). Therefore, the volume is \( 13.32 \times 10^{17} \text{ m}^3. \)

Step 4: Convert to Scientific Notation If Needed

Write \( 13.32 \times 10^{17} \) in a proper scientific notation format: \( 1.332 \times 10^{18} \text{ m}^3 \).

Step 5: Convert Cubic Meters to Liters

Since 1 cubic meter equals 1000 liters, multiply the volume in cubic meters by 1000 to convert to liters: \[ 1.332 \times 10^{18} \times 10^3 = 1.332 \times 10^{21} \text{ liters}. \]

Key Concepts

Scientific Notation

Scientific notation is a way to express very large or very small numbers in a compact form. This is especially helpful in disciplines like oceanography, where you often encounter measurements of significant magnitude. It is expressed as the product of a number between 1 and 10 and a power of 10.

For example, in our exercise, the depth of the ocean is given as \(3.7 \times 10^3\) meters. This means \(3.7 \times 10^3\) is equal to \(3,700\) meters. Similarly, the area of the ocean is \(3.6 \times 10^{14}\) square meters, equal to \(360,000,000,000,000\) square meters.

• The number always consists of:
  • A coefficient: A decimal number between 1 and 10 (e.g., 3.7).
  • An exponent: An integer that tells you how many times to multiply the coefficient by 10 (e.g., \(10^3\)).
• This notation simplifies calculations and conveys the scale of quantities naturally and efficiently.

Understanding scientific notation is crucial when calculating volumes of features like oceans, given their vast areas and depths.

Unit Conversion

Unit conversion is an essential skill that allows you to translate between different units of measurement, ensuring accuracy in calculations. When calculating the ocean's volume, you start with dimensions in cubic meters and often need to convert to more familiar units, such as liters.

In this exercise, the volume of the ocean is converted from cubic meters to liters. Since 1 cubic meter is equivalent to 1000 liters, the conversion involves multiplying the volume in cubic meters by 1000.

• Formula: \[ \, \text{Liters} = \text{Cubic Meters} \times 1000 \, \]
• The initial volume in cubic meters was \(1.332 \times 10^{18} \text{ m}^3\), and after multiplication by \(10^3\) (which is 1000), it becomes \(1.332 \times 10^{21}\) liters.

This kind of conversion is vital, as it not only makes the data more comprehensible but also aligns with everyday measurement units.

Oceanography Volumes

Calculating the volume of the oceans is an example of the application of volume formulas in oceanography. The volume is determined by multiplying the surface area by the average depth, as shown by the formula: \( \text{Volume} = \text{Area} \times \text{Depth} \).

Oceans cover vast areas and have substantial depths, making scientific notation and precise calculations necessary. Here, the area is \(3.6 \times 10^{14} \text{ m}^2\) and the depth is \(3.7 \times 10^3 \text{ m}\).

• Multiplying these gives: \[ \, \text{Volume} = (3.6 \times 10^{14}) \times (3.7 \times 10^3) = 13.32 \times 10^{17} \text{ m}^3 \, \]
• This is then rewritten in scientific notation as \(1.332 \times 10^{18} \text{ m}^3\).
• Finally, converting this to liters (as discussed earlier) concludes the conversion process, yielding a final volume of \(1.332 \times 10^{21}\) liters.

These calculations are important not just theoretically but also practically, affecting fields like climate science and marine resource management.