Problem 30 - General Chemistry: Principles and Modern Applications

Question

Determine the number of the following: (a) square meters \(\left(\mathrm{m}^{2}\right)\) in 1 square kilometer \(\left(\mathrm{km}^{2}\right)\) (b) cubic centimeters \(\left(\mathrm{cm}^{3}\right)\) in 1 cubic meter \(\left(\mathrm{m}^{3}\right)\) (c) square meters \(\left(\mathrm{m}^{2}\right)\) in 1 square mile \(\left(\mathrm{mi}^{2}\right)\) \((1 \mathrm{mi}=5280 \mathrm{ft})\)

Step-by-Step Solution

Answer

(a) There are 1,000,000 square meters in 1 square kilometer. (b) There are 1,000,000 cubic centimeters in 1 cubic meter. (c) There are approximately 2,589,988.11 square meters in 1 square mile.

Step 1: Convert square kilometers to square meters

A square kilometer is a unit of area equivalent to a square with sides of 1 kilometer. As a kilometer is equal to 1000 meters, we square this conversion factor to find that 1 square kilometer equals \(1000^2 = 1,000,000\) square meters.

Step 2: Convert cubic meters to cubic centimeters

A cubic meter is a unit of volume equivalent to a cube with sides of 1 meter. As a meter is equal to 100 centimeters, we cube this conversion factor to find that 1 cubic meter equals \(100^3 = 1,000,000\) cubic centimeters.

Step 3: Convert square miles to square meters

A square mile is a unit of area equivalent to a square with sides of 1 mile. As a mile is equal to 5280 feet and 1 foot is equal to 0.3048 meters, we square this conversion factor to find that 1 square mile equals \((5280 * 0.3048)^2 = 2,589,988.110336\) square meters.

Key Concepts

Square Kilometer to Square Meter

Converting from square kilometers to square meters is a straightforward process. Imagine a square that has sides measuring 1 kilometer each. A kilometer equals 1,000 meters, so when you want to find the area in square meters, you simply convert each side's length to meters and then calculate the area:

• 1 kilometer is 1,000 meters.
• There are 1,000 meters in each side, and since area is length squared, you multiply 1,000 by 1,000.
• This gives you: \(1,000 \times 1,000 = 1,000,000\) square meters.

This conversion tells us that a square kilometer is exactly 1,000,000 square meters. Remembering that 1 square kilometer equals 1,000,000 square meters can be particularly handy when you're working with large land areas.

Cubic Meter to Cubic Centimeter

Converting cubic meters to cubic centimeters involves understanding volume conversion. A cubic meter is essentially a cube with each side measuring 1 meter. To convert to cubic centimeters, consider the conversion factor between meters and centimeters:

• 1 meter equals 100 centimeters.
• For volume, you cube the conversion factor, because you're considering three dimensions (length, width, height).
• Thus, you calculate \(100^3\) which equals 1,000,000.

So, 1 cubic meter is equivalent to 1,000,000 cubic centimeters. This is useful for smaller volumes or when dealing with scientific measurements where cubic centimeters are more commonly used.

Square Mile to Square Meter

Square miles to square meters conversion requires a bit more calculation, primarily since you're working with two different unit systems: miles and meters. Here's how to tackle it:

• 1 mile is 5,280 feet.
• Each foot is converted to meters: 1 foot equals 0.3048 meters.
• To find the area in square meters, calculate \((5,280 \times 0.3048)^2\).
• This results in exactly 2,589,988.110336 square meters per square mile.

This conversion is especially important in geography and real estate, where large areas need to be converted from imperial to metric units. Remembering these detailed steps can help simplify your calculations.