Chapter 2 Discussion/Participation

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Using & Understanding Mathematics: A Quantitative Reasoning Approach

Eighth Edition

Chapter 2

Approaches to Problem Solving

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Unit 2B Extending Unit Analysis

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U.S. Customary System

The U.S. customary system has roots dating back thousands of years, and its units became standardized in often surprising ways. For example, the modern length of 1 yard was defined by English King Henry I (1100–1135), who decreed it to be the distance from the tip of his nose to the tip of his thumb on his outstretched arm.

The table in your text summarizes the official U.S. customary system, showing standard units for length, weight, and volume.

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Example: The Kentucky Derby (1 of 2)

The length of the Kentucky Derby horse race is 10 furlongs. How long is the race in miles?

Solution

See page 78 for Table 2.1,

which is the same as 0.125 mile. We can write the conversion factor in two other equivalent forms:

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Example: The Kentucky Derby (2 of 2)

The length of the Kentucky Derby horse race is 10 furlongs. How long is the race in miles?

The Kentucky Derby is a race of 1.25 miles.

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Example: Price Comparison (1 of 2)

You are planning to make pesto and need to buy basil. At the grocery store, you can buy small containers of

basil priced at $2.99 for each

-ounce container.

At the farmer’s market, you can buy basil in bunches for $12 per pound. Which is the better deal?

Solution

To compare the prices, we need them both in the same units.

Convert the small container price to a price per pound.

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Example: Price Comparison (2 of 2)

The container price is

ounce, which

means we need to divide. We then multiply by the conversion of 16 ounces per pound:

The small containers are priced at almost $72 per pound, which is six times as much as the farmer’s market price.

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Metric System

The international metric system was invented in France late in the 18th century for two primary reasons: (1) to replace many customary units with just a few basic units and (2) to simplify conversions through use of a decimal (base 10) system. The basic units of length, mass, time, and volume in the metric system are

the meter for length, abbreviated m

the kilogram for mass, abbreviated kg

the second for time, abbreviated s

the liter for volume, abbreviated L

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Common Metric Prefixes

Small Values

Large Values

* Micro is usually abbreviated with µ (Greek letter mu), but in medical fields it is common to use “mc” intead.

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Example: Using Metric Prefixes

Convert 2759 centimeters to meters.

Solution

The table shows that centi means

or, equivalently, 1 m = 100 cm. Therefore, 2759 centimeters is the same as

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Example: Marathon Distance

The marathon running race is about 26.2 miles. About how far is it in kilometers?

Solution

Table 2.3 shows that 1 mi = 1.6093 km. We use the conversion in the form with miles in the denominator to find

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Temperature Units (1 of 3)

The Fahrenheit scale, commonly used in the United States, is defined so water freezes at 32°F and boils at 212°F.

The rest of the world uses the Celsius scale, which places the freezing point of water at 0°C and the boiling point at 100°C.

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Temperature Units (2 of 3)

In science, we use the Kelvin scale, which is the same as the Celsius scale except for its zero point, which corresponds to −°273.15°C. A temperature of 0 K is known as absolute zero, because it is the coldest possible temperature. (The degree symbol

is not used on the Kelvin scale.)

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Temperature Units (3 of 3)

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Temperature Conversions

The conversions are given both in words and with formulas in which C, F, and K are Celsius, Fahrenheit, and Kelvin temperatures, respectively.

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Example: Human Body Temperature

Average human body temperature is 98.6°F. What is it in Celsius and Kelvin?

Solution

Convert from Fahrenheit to Celsius by subtracting 32 and then dividing by 1.8:

We find the Kelvin equivalent by adding 273.15 to the Celsius temperature:

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Units of Energy and Power

Energy is what makes matter move or heat up. International metric unit is the joule.

Power is the rate at which energy is used. International metric unit is the watt.

A kilowatt-hour is a unit of energy.

1 kilowatt-hour = 3.6 million joules

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Example: Operating Cost of a Light Bulb (1 of 2)

A utility company charges

per kilowatt-hour of

electricity. How much does it cost to keep a 100-watt light bulb on for a week? How much will you save in a year if you replace the bulb with an L E D bulb that provides the same amount of light for only 25 watts of power?

Solution

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Example: Operating Cost of a Light Bulb (2 of 2)

Now find the cost.

The electricity for the bulb costs $2.52 per week. If you replace the 100-watt bulb with a 25-watt L E D, you’ll use only

as much energy, which means your weekly cost will be

only

In other words, your savings will be $2.52 − $0.63

= $1.89 per week, so in a year you’ll save about:

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Units of Density and Concentration (1 of 2)

Density describes compactness or crowding.

Material density is given in units of mass per unit volume.

e.g., grams per cubic centimeter

Population density is given by the number of people per unit area.

e.g., people per square mile

Information density is given in units of mass per unit volume.

e.g., gigabytes per square inch

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Units of Density and Concentration (2 of 2)

Concentration describes the amount of one substance mixed with another.

The concentration of an air pollutant is often measured by the number of molecules of the pollutant per million molecules of air.

e.g., parts per million (p p m)

Blood alcohol content (B A C) describes the concentration of alcohol in a person’s body.

e.g., grams of alcohol per 100 milliliters of blood

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Example: Ear Infection (1 of 3)

A child weighing 15 kilograms has a bacterial ear infection. A physician orders treatment with amoxicillin at a dosage based on 30 milligrams per kilogram of body weight per day, divided into doses every 12 hours.

a. How much amoxicillin should the child be prescribed every 12 hours?

b. If the medicine is to be taken in a liquid suspension with concentration 25 mg/ml, how much should the child take every 12 hours?

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Example: Ear Infection (2 of 3)

a. How much amoxicillin should the child be prescribed every 12 hours?

The prescribed dosage is 30 mg/kg of body weight per day, but because it will be given in two doses (every 12 hours), each dose will be based on half of the total, or 15 mg/kg of body weight. Therefore, for a child weighing 15 kilograms, the dosage should be

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Example: Ear Infection (3 of 3)

b. If the medicine is to be taken in a liquid suspension with concentration 25 mg/ml, how much should the child take every 12 hours?

The liquid suspension contains 25 milligrams of amoxicillin per milliliter (ml) of liquid, and from part (a) we know the total amount of amoxicillin in each dose should be 225 mg. We are looking for the total amount of liquid that the child should be given for each dose, so the answer should have units of milliliters.

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Chapter 2 Discussion/Participation

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