Problem 89 - General Chemistry

Question

The first measurement of sea depth was made in 1840 in the central South Atlantic, where a plummet was lowered 2425 fathoms. What is this depth in meters? Note that 1 fathom \(=6 \mathrm{ft}, 1 \mathrm{ft}=12 \mathrm{in} .\), and \(1 \mathrm{in.}=2.54 \times 10^{-2} \mathrm{~m}\). (These relations are exact.)

Step-by-Step Solution

Answer

The depth is 4434.84 meters.

Step 1: Convert Fathoms to Feet

First, we need to convert the sea depth from fathoms to feet. Given that 1 fathom equals 6 feet, we can multiply the total fathoms by 6 to find the depth in feet. \[ 2425 \text{ fathoms} \times 6 \text{ feet per fathom} = 14550 \text{ feet} \]

Step 2: Convert Feet to Inches

Next, convert the depth from feet to inches. Since 1 foot equals 12 inches, multiply the depth in feet by 12.\[ 14550 \text{ feet} \times 12 \text{ inches per foot} = 174600 \text{ inches} \]

Step 3: Convert Inches to Meters

Finally, convert the depth from inches to meters. We know that 1 inch is exactly equal to 0.0254 meters. Hence, multiply the total inches by 0.0254 to find the depth in meters.\[ 174600 \text{ inches} \times 0.0254 \text{ meters per inch} = 4434.84 \text{ meters} \]

Key Concepts

Fathoms to Meters

When measuring sea depth, fathoms are commonly used. A fathom is a unit of measurement that traditionally equates to six feet in length. To convert fathoms to meters involves several steps:
1. **Convert fathoms to feet**: Multiply the number of fathoms by 6, since there are six feet in one fathom. For example, converting 2425 fathoms to feet gives 14550 feet. 2. **Convert feet to inches**: Since every foot is equivalent to 12 inches, multiply the number of feet by 12. This step transforms 14550 feet into 174600 inches.
3. **Convert inches to meters**: Utilize the conversion factor where 1 inch equals 0.0254 meters. By multiplying your total inches by this number, you complete the conversion from fathoms to meters. In our example, 174600 inches convert to 4434.84 meters.
By following these steps, you can accurately convert any measurement given in fathoms to meters, a standard metric unit.

Measurement Conversions

Measurement conversions allow us to switch between different units of measurement, an essential skill in both academics and real-world applications. These conversions typically follow a step-by-step approach to ensure accuracy. Here’s how you can approach them effectively:

• Identify your starting unit and the final desired unit.
• Understand the relationship between the units through conversion factors. Each conversion involves a multiplication or division that corresponds to these factors.
• Execute the conversion one step at a time rather than trying to jump directly from the starting unit to the final unit. This helps to minimize errors.

In these conversions, it's crucial to work systematically. Always double-check your math, especially when dealing with small decimals or fractions, as they can significantly impact the final result.

Metric System

The metric system is an international system of measurement that is decimal-based and easy to convert. It uses units such as meters for length, kilograms for mass, and liters for volume. Here are some of its notable features:

• **Uniformity**: The system is consistent with its use of powers of ten, making it easy to scale measurements up or down.
• **Global Acceptance**: The metric system is widely used around the world, making international collaboration and communication straightforward.
• **Base Units**: Each type of measurement is anchored by a base unit. For example, the meter is the fundamental unit for measuring length within the system.

Conversions within the metric system often involve shifting the decimal point. This simplicity is one of the reasons the metric system is preferred for scientific and many day-to-day uses. Understanding and being comfortable converting within the metric system can significantly enhance your mathematical and scientific interactions.