Question

1. metric conversions. what are the steps to metric conversions? eg 35 mm to cm to m to km. define the prefixes milli-, centi-, and kilo-.

Explanation:

1. Steps for Metric Conversions (using 35 mm to cm to m to km as example):

• Step 1: Understand Metric Prefix Relationships

The metric system uses prefixes that represent powers of 10. The base unit for length is the meter (m). The relationships between the prefixes for length (and most metric units) are:

• 1 meter (\(m\)) = 100 centimeters (\(cm\)) (since "centi-" means \(10^{-2}\))
• 1 meter (\(m\)) = 1000 millimeters (\(mm\)) (since "milli-" means \(10^{-3}\))
• 1 kilometer (\(km\)) = 1000 meters (\(m\)) (since "kilo-" means \(10^{3}\))

• Step 2: Convert 35 mm to cm

To convert millimeters to centimeters, use the conversion factor \(1\ cm = 10\ mm\) (or \(1\ mm = \frac{1}{10}\ cm\)).
\(35\ mm \times \frac{1\ cm}{10\ mm} = 3.5\ cm\) (the "mm" units cancel out, leaving "cm").

• Step 3: Convert 3.5 cm to m

Use the conversion factor \(1\ m = 100\ cm\) (or \(1\ cm = \frac{1}{100}\ m\)).
\(3.5\ cm \times \frac{1\ m}{100\ cm} = 0.035\ m\) (the "cm" units cancel out, leaving "m").

• Step 4: Convert 0.035 m to km

Use the conversion factor \(1\ km = 1000\ m\) (or \(1\ m = \frac{1}{1000}\ km\)).
\(0.035\ m \times \frac{1\ km}{1000\ m} = 3.5 \times 10^{-5}\ km\) (the "m" units cancel out, leaving "km").

1. Define the Prefixes:

• Milli- (\(\boldsymbol{m}\)): A metric prefix meaning one-thousandth (\(10^{-3}\)) of the base unit. For example, 1 millimeter (\(mm\)) = \(10^{-3}\) meters, 1 milligram (\(mg\)) = \(10^{-3}\) grams.
• Centi- (\(\boldsymbol{c}\)): A metric prefix meaning one-hundredth (\(10^{-2}\)) of the base unit. For example, 1 centimeter (\(cm\)) = \(10^{-2}\) meters, 1 centiliter (\(cL\)) = \(10^{-2}\) liters.
• Kilo- (\(\boldsymbol{k}\)): A metric prefix meaning one-thousand (\(10^{3}\)) times the base unit. For example, 1 kilometer (\(km\)) = \(10^{3}\) meters, 1 kilogram (\(kg\)) = \(10^{3}\) grams.

Answer:

(for the example conversion):

• \(35\ mm = 3.5\ cm\)
• \(3.5\ cm = 0.035\ m\)
• \(0.035\ m = 3.5 \times 10^{-5}\ km\)

(For the prefix definitions, see the explanations above.)