Chapter 2: Graphing & Geometry Section 2.2: The Metric System & Dimensional Analysis Applied Mathematics, 2nd Ed, Copyright 2018, Matovina & Yates

Applied Mathematics, 2nd Ed, Copyright 2018, Matovina & Yates The Metric System According to Wikipedia, the United States is the only industrialized country that has not adopted the metric system as its official system of measurement. Why not? The Metric System is easier to learn and easier to use. If you are trying to remove a standard bolt and find a 1/2-inch wrench is too small, but a 5/8-inch wrench is too big, what size is between those measures? If you are trying to remove a metric bolt and find an 11-mm wrench is too small, but a 13-mm wrench is too big, what size metric wrench is between those measures? So why haven’t we switched? Americans are stubborn; they don’t want to relearn their height in centimeters and their weight in kilograms. Applied Mathematics, 2nd Ed, Copyright 2018, Matovina & Yates

Applied Mathematics, 2nd Ed, Copyright 2018, Matovina & Yates The Metric System Historical Measures: In early times, there was not a need for precise measurements, so approximations were used. Foot: The length of the adult human foot. Yard: The distance from the tip of the nose to the end of an outstretched arm. Inch: The length of three barley corns laid end-to-end. Fathom: The length of a full arm span. Acre: The amount of land a horse could plow in a day. Applied Mathematics, 2nd Ed, Copyright 2018, Matovina & Yates

Applied Mathematics, 2nd Ed, Copyright 2018, Matovina & Yates The Metric System In the 1700s, the French Government devised a new system of measure that was easily portable, convertible, and interrelated – the metric system. This new system was based entirely on the meter, which was defined as one ten-millionth of the distance from the equator to the North Pole along the line of longitude passing through Paris. Then, they scaled it up and down according to multiples of 10, giving each multiple a corresponding prefix. For example, a centimeter (cm) was defined as 1/100 of a meter. Then, they stated the meter (m) would measure length, the liter (L) would measure volume, and the gram (g) would measure mass. Finally, the sea-level equivalency was made: 1 cm3 = 1 mL = 1 g Applied Mathematics, 2nd Ed, Copyright 2018, Matovina & Yates

The Metric System Base Unit: meter (m), gram (g), liter (L) Prefix (abbreviation) Multiple Example mega- (M-) 106 = 1,000,000 megameter (Mm), megabucks kilo- (k-) 103 = 1,000 kilogram (kg) hecto- (h-) 102 = 100 hectoliter (hL) deka- or deca- (da-) 101 = 10 decameter (dam) Base Unit: meter (m), gram (g), liter (L) deci- (d-) 10-1 = 1/10 = 0.1 decimeter (dm) centi- (c-) 10-2 = 1/100 = 0.01 centimeter (cm) milli- (m-) 10-3 = 1/1000 = 0.001 milligram (mg) micro- ( 10-6 = 1/1,000,000 = 0.0000001 microgram (𝜇g or mcg) 𝜇- or mc-) Applied Mathematics, 2nd Ed, Copyright 2018, Matovina & Yates

Applied Mathematics, 2nd Ed, Copyright 2018, Matovina & Yates The Metric System Portable: Travelers could carry a meter with them. Convertible: With the prefixes, one could easily convert from one unit to another by simply moving the decimal point. Interrelated: Picture a cubic centimeter filled with water at sea level. 1 cm 1 cm3 = 1 mL = 1 g Applied Mathematics, 2nd Ed, Copyright 2018, Matovina & Yates

Applied Mathematics, 2nd Ed, Copyright 2018, Matovina & Yates The Metric System As a frame of reference, here are some basic comparisons: A meter is slightly longer than a yard. Your little finger is about one centimeter wide. A dime is one millimeter thick. A kilometer is a little more than a half-mile. A dollar bill weighs about one gram. A nickel weighs exactly 5 grams. A kilogram is about the weight of a large tub of butter (2.2 pounds). A liter is a little more than a quart. A milliliter is about a drop. A kiloliter is over 260 gallons. Applied Mathematics, 2nd Ed, Copyright 2018, Matovina & Yates

Applied Mathematics, 2nd Ed, Copyright 2018, Matovina & Yates The Metric System To convert between metric measures, simply move the decimal point in the same number places as the difference in the order of the units – noting that, when the unit gets smaller the quantity must get larger, and when the unit gets larger, the quantity must get smaller. Example: Change 23 mm to cm. From mm to cm is one unit, so the decimal point must be moved one place, meaning the quantity will either be 2.3 or 230. Since a cm is larger than an mm, make the quantity smaller. 23 mm = 2.3 cm Applied Mathematics, 2nd Ed, Copyright 2018, Matovina & Yates

Applied Mathematics, 2nd Ed, Copyright 2018, Matovina & Yates Dimensional Analysis Dimensional analysis is just that – analyzing the dimensions in a conversion. The process involves using unit fractions, which is a fraction consisting of two equal measurements. Examples of unit fractions include: 1 𝑚 100 𝑐𝑚 , 100 𝑐𝑚 1 𝑚 , 60 𝑚𝑖𝑛 1 ℎ𝑟 , 1 𝑦𝑑 3 𝑓𝑡 , 1 𝑚𝑖 5280 𝑓𝑡 Applied Mathematics, 2nd Ed, Copyright 2018, Matovina & Yates

Applied Mathematics, 2nd Ed, Copyright 2018, Matovina & Yates Dimensional Analysis To convert between measures using dimensional analysis, multiply the starting measure by a unit fraction that will force the unwanted units to cancel, while leaving the desired units. Then multiply or divide the numbers, accordingly. Example: Convert 3 days to hours. 3 days = 3 𝑑𝑎𝑦𝑠 1 × 24 ℎ𝑟 1 𝑑𝑎𝑦 = 3×24 ℎ𝑟 1 = 72 hr Applied Mathematics, 2nd Ed, Copyright 2018, Matovina & Yates

Applied Mathematics, 2nd Ed, Copyright 2018, Matovina & Yates Dimensional Analysis Example: Convert 30 ft/sec to mi/hr. Round the answer to the nearest hundredth. 30 ft/sec = 30 𝑓𝑡 1 𝑠𝑒𝑐 × 1 𝑚𝑖 5280 𝑓𝑡 × 60 𝑠𝑒𝑐 1 𝑚𝑖𝑛 × 60 𝑚𝑖𝑛 1 ℎ𝑟 = 30 × 60 × 60 𝑚𝑖 5280 ℎ𝑟 = 20.45 mph Applied Mathematics, 2nd Ed, Copyright 2018, Matovina & Yates