Question

A 6.0 -ft person weighs 168 lb. Express this person's height in meters and weight in kilograms \((1 \mathrm{lb}=453.6 \mathrm{~g} ;\) \(1 \mathrm{~m}=3.28 \mathrm{ft} \mathrm{t}\).

Short answer

Height is 1.83 meters and weight is 76.20 kilograms.

Step-by-step solution

Convert Height from Feet to Meters

To convert height from feet to meters, use the conversion factor, where 1 meter equals 3.28 feet. Given the person's height is 6.0 feet, the conversion is calculated as follows: \[ \text{Height in meters} = \frac{6.0 \text{ ft}}{3.28 \text{ ft/m}} \approx 1.83 \text{ m} \].

Convert Weight from Pounds to Kilograms

To convert weight from pounds to kilograms, use the conversion factor, where 1 pound equals 453.6 grams. Since 1000 grams equals 1 kilogram, we calculate the person's weight in kilograms by first converting pounds to grams, then grams to kilograms: \[ \text{Weight in kilograms} = \frac{168 \text{ lb} \times 453.6 \text{ g/lb}}{1000 \text{ g/kg}} = 76.2048 \text{ kg} \].

Key concepts

Conversion Factors

When dealing with unit conversions in chemistry, conversion factors are crucial tools. They are ratios derived from the equality between different units and allow us to convert a quantity from one unit to another. For example, in the exercise above, two main conversion factors are used: - 1 meter = 3.28 feet - 1 pound = 453.6 grams

To use a conversion factor, we set it up as a fraction that equals one, either as \( \frac{1 \, \text{m}}{3.28 \, \text{ft}} \) or \( \frac{3.28 \, \text{ft}}{1 \, \text{m}} \), depending on which direction you're converting. Properly selecting the conversion factor ensures that the units you start with cancel out, leaving you with the desired units. For instance, when converting feet to meters, using \( \frac{1 \, \text{m}}{3.28 \, \text{ft}} \) cancels out the feet units effectively.

Always keep in mind that a correct conversion factor maintains the integrity of the quantity, meaning the actual amount of the physical quantity doesn't change, only the unit expression does.

Metric System

The metric system is an international system of measurements based on powers of ten, making conversions straightforward. It is extensively used worldwide in scientific and everyday contexts, providing a unified method for measuring various quantities like length, mass, and volume.

Units of measurement in the metric system include:

• The meter (m) for length
• The kilogram (kg) for mass
• The liter (L) for volume

This system simplifies conversions as it relies on multiplying or dividing by ten, a base ten system. In our exercise, we convert feet to meters using the metric system. One meter is approximately 3.28 feet. By dividing the length in feet by 3.28, we seamlessly convert this measurement to meters.

The straightforward nature of the metric system makes it a preferred choice in scientific research and international trade.

Dimensional Analysis

Dimensional analysis is a systematic approach to converting between units by employing conversion factors. This method ensures the proper cancellation of units throughout the calculation process, maintaining the consistency of the physical quantity you calculate.

Using dimensional analysis, you set up a problem with the initial measurement and then apply conversion factors as a series of fractions, where units diagonally opposite each other cancel out.

For instance: \[ \text{Height in meters} = 6.0 \, \text{ft} \times \frac{1 \, \text{m}}{3.28 \, \text{ft}} \] Here, the feet units cancel, leaving the result in meters.

By using dimensional analysis to solve these kinds of problems, you can ensure clarity and accuracy without having to memorize conversion formulas. The method is universally applicable across diverse physical quantities.

Weight and Mass Conversion

Weight and mass are physical quantities often converted in scientific applications. Weight refers to the force exerted by gravity on an object's mass, while mass is the amount of matter in an object.

In the exercise, the conversion from pounds to kilograms involves first understanding that the weight in pounds must be converted to mass in grams, then to kilograms. - 1 pound is equivalent to 453.6 grams - 1 kilogram is 1000 grams
Thus, for 168 pounds: \[ \text{Mass in kilograms} = 168 \, \text{lb} \times \frac{453.6 \, \text{g}}{\text{lb}} \times \frac{1 \, \text{kg}}{1000 \, \text{g}} \] By following this calculation, 168 pounds converts to approximately 76.20 kilograms.

Understanding these conversions is valuable not just in classroom settings but also in real-world contexts where unit conversions are frequently necessary, such as in medicine and engineering.