Problem 66 - Chemistry

Question

Carry out the following conversions: (a) How many grams of meat are in a quarter-pound hamburger \((0.25 \mathrm{lb}) ?\) (b) How tall in meters is the Willis Tower, formerly called the Sears Tower, in Chicago ( \(1454 \mathrm{ft}\) )? (c) How large in square meters is the land area of Australia \(\left(2,941,526 \mathrm{mi}^{2}\right) ?\)

Step-by-Step Solution

Answer

(a) 113.4 g; (b) 443.18 m; (c) 7,615,752,117,000 m^2.

Step 1: Understand Conversion Factors for Weight

To convert pounds (lb) to grams (g), we use the conversion factor \(1 \text{ lb} = 453.59237 \text{ g}\). This allows us to transform the weight measurement from the imperial system to the metric system.

Step 2: Convert Pounds to Grams

Using the conversion factor, calculate: \[ 0.25 \text{ lb} \times 453.59237 \text{ g/lb} = 113.3980925 \text{ g} \] Hence, there are approximately \(113.4 \text{ g}\) of meat in the hamburger.

Step 3: Understand Conversion Factors for Length

To convert feet (ft) to meters (m), we use the conversion factor \(1 \text{ ft} = 0.3048 \text{ m}\). This is required for changing the unit of height measurement from feet to meters.

Step 4: Convert Feet to Meters

Using the foot to meter conversion factor, calculate: \[ 1454 \text{ ft} \times 0.3048 \text{ m/ft} = 443.1762 \text{ m} \] Therefore, the Willis Tower is approximately \(443.18 \text{ m}\) tall.

Step 5: Understand Conversion Factors for Area

For converting square miles (mi\(^2\)) to square meters (m\(^2\)), note that \(1 \text{ mi} = 1609.34 \text{ m}\). Thus, \(1 \text{ mi}^2 = (1609.34 \text{ m})^2 = 2,589,988.11 \text{ m}^2\).

Step 6: Convert Square Miles to Square Meters

Using the area conversion factor, calculate: \[ 2,941,526 \text{ mi}^2 \times 2,589,988.11 \text{ m}^2/ ext{mi}^2 = 7,615,752,116,696.61 \text{ m}^2 \] Therefore, the land area of Australia is approximately \(7,615,752,117,000 \text{ m}^2\).

Key Concepts

Conversion Factors

Conversion factors are crucial tools in mathematics and science that help us change measurements from one unit to another. They are essentially ratios or fractions that equal 1, allowing the measurement not to change in value, only in units.
For instance, to convert weight from pounds to grams, we use the conversion factor:

• 1 pound (lb) is equal to 453.59237 grams (g).

The conversion factor works by multiplying the original measurement by the ratio of the desired unit over the original unit, ensuring the original value is not altered, just expressed differently. Understanding significant conversion factors like these is key when transitioning measurements from imperial to metric systems or vice versa.

Imperial to Metric

The imperial and metric systems are two main measurement systems used worldwide. The imperial system, primarily used in the United States, includes units like inches, feet, and pounds. In contrast, the metric system, used globally and especially in scientific contexts, includes units like meters, liters, and grams.
When converting from imperial to metric, knowing the precise conversion factors is essential. For height, for example:

• 1 foot equals 0.3048 meters.

And for area:

• 1 square mile (mi²) equals 2,589,988.11 square meters (m²).

These conversions allow us to easily convert measurements from one system to another, maintaining accuracy and consistency.

Weight Conversion

Weight conversion often involves translating pounds into grams, kilograms, or other units. When we convert a quarter-pound (0.25 lb) hamburger into grams, we want to use a reliable conversion factor:

• 1 pound (lb) equals 453.59237 grams (g).

Thus, by multiplying 0.25 pound by 453.59237 grams per pound, you see that the hamburger is about 113.4 grams in weight.
Weight conversion calculations like this are vital when dealing with food, nutrition, and chemical quantities, ensuring measurement precision across different unit systems.

Length Conversion

Length conversion is commonly required when dealing with measurements of height, distance, or dimensions from one unit system to another. To convert the height of the Willis Tower from feet to meters, use the conversion factor:

• 1 foot (ft) is approximately equal to 0.3048 meters (m).

Multiplying the height in feet, 1454 ft, by 0.3048 gives 443.18 meters.
Grasping length conversion is essential in fields like engineering, architecture, and scientific research where accurate spatial dimensions are necessary.

Area Conversion

Area conversion is used to translate the measure of space an object covers, from one unit to another. To find the land area of Australia in square meters, we start by knowing the critical conversion factor:

• 1 square mile (mi²) equals 2,589,988.11 square meters (m²).

By using this factor, we can convert 2,941,526 mi² (Australia's land area) to the metric equivalent.
Calculating this gives approximately 7,615,752,117,000 m². Area conversion proves essential in geography, real estate, and environmental sciences, providing a common ground for assessing land sizes accurately worldwide.