Question

Use metric conversion factors to solve each of the following problems: a. The height of a student is \(175 \mathrm{~cm}\). How tall is the student in meters? b. A cooler has a volume of \(5500 \mathrm{~mL}\). What is the capacity of the cooler in liters? c. A Bee Hummingbird has a mass of \(0.0018 \mathrm{~kg}\). What is the mass of the hummingbird in grams?

Short answer

a. 1.75 meters; b. 5.5 liters; c. 1.8 grams.

Step-by-step solution

- Convert height from centimeters to meters

To convert from centimeters to meters, use the conversion factor: 1 meter = 100 centimeters Therefore, \[175 \text{ cm} = \frac{175}{100} \text{ m} = 1.75 \text{ m}\]

- Convert volume from milliliters to liters

To convert from milliliters to liters, use the conversion factor: 1 liter = 1000 milliliters Therefore, \[5500 \text{ mL} = \frac{5500}{1000} \text{ L} = 5.5 \text{ L}\]

- Convert mass from kilograms to grams

To convert from kilograms to grams, use the conversion factor: 1 kilogram = 1000 grams Therefore, \[0.0018 \text{ kg} = 0.0018 \times 1000 \text{ g} = 1.8 \text{ g}\]

Key concepts

Unit Conversion

Unit conversion is the process of changing a measure of physical quantity from one unit to another. This is crucial in ensuring accurate communication of measurements.
For example, converting centimeters to meters, milliliters to liters, or kilograms to grams requires using appropriate conversion factors. A conversion factor is a number used to change one set of units to another, by multiplying or dividing.

Unit conversion is particularly important in science, engineering, and daily life where precise measurements are essential.
Here are some common conversion factors used in the metric system:

• 1 meter = 100 centimeters
• 1 liter = 1000 milliliters
• 1 kilogram = 1000 grams

Metric System

The metric system is an international system of measurement based on powers of 10. This system uses base units like meters for length, liters for volume, and grams for mass, which can be scaled up or down by multiplying or dividing by 10, 100, 1000, etc.
The simplicity and uniformity of the metric system make it widely used globally, especially in scientific contexts.

Key features of the metric system include:

• Standardization: The same units are used worldwide, reducing confusion.
• Scalability: Units are easily scaled by multiples of 10, making conversions straightforward.
• Consistency: All derived units (like area, volume) stem from base units.

A practical example of the metric system's simplicity is converting 175 centimeters to 1.75 meters by dividing by 100, as there's a consistent conversion factor between units.

Dimensional Analysis

Dimensional analysis, or unit analysis, is a method used to convert one kind of unit to another and to ensure equations are dimensionally consistent. It involves multiplying the quantity to be converted by one or more conversion factors expressed as ratios that equal 1.
For instance, converting 5500 milliliters to liters involves the conversion factor ratio of 1 liter = 1000 milliliters, so:
\[ 5500 \text{ mL} = \frac{5500}{1000} \text{ L} = 5.5 \text{ L} \]
Here, the units of milliliters cancel out, leaving the result in liters.
Dimensional analysis is invaluable in verifying complex scientific equations and calculations, preventing unit-related errors, and providing a systematic approach to problem-solving.
An interesting feature of dimensional analysis is its ability to handle multiple conversions in one step.
For example, converting 0.0018 kg to grams directly by multiplying by 1000:
\[ 0.0018 \text{ kg} = 0.0018 \times 1000 \text{ g} = 1.8 \text{ g} \]