VWT 272
Week 2
Quiz 1
Number of quizzes taken
Min
Max
Mean
Median
Mode
22
14
30
28.3
30
30
Week 2
Dimensional Analysis
The BEST Trick EVER
“Believe nothing merely because you
have been told it. Do not believe
what your teacher tells you merely
out of respect for the teacher. But
whatsoever, after due examination
and analysis, you find to be kind,
conducive to the good, the benefit,
the welfare of all beings -- that
doctrine believe and cling to, and
take it as your guide.”
- Buddha (c 563 – 483 BCE)
Plan of Study
• Dimensional Analysis
Dimensional Analysis
• Definitions (from the web)
– A method of using the known units in a problem to help
deduce the process of arriving at a solution (about.com)
– A technique of calculating with physical quantities in which
units are included and treated in the same way as numbers
(dictionary.com)
– A problem-solving method that uses the fact that any number
or expression can be multiplied by one without changing its
value (chem.tamu.edu)
– is by far the most useful math trick you'll ever learn
(alysion.org)
Dimensional Analysis
• Limitations
– Useful for measured things
– Can NOT be used with exponential or trigonometric functions
Dimensional Analysis
• Based on 5 Properties of Multiplication & 1 of Division
– Commutative Property
• 4x2=2x4
– Associative Property
• (2 x 3) x 4 = 2 x (3 x 4)
– Distributive Property
• 2 x (3 + 4) = (2 x 3) + (2 x 4)
– Multiplicative Identity Property
• 4x1=4
– Divisive Identity Property
• 4 / 1 = 4 or 4/4 = 1
– Multiplicative Inverse Property
• 4 / 2 = 4 x 1/2
Dimensional Analysis
• Start with lots of room to work
• Figure out what the question really is about
– decide what is and what is not useful
•
•
•
•
•
List what you know (include units)
List what you want to know (include units)
Feel free to break the question into logical parts
Draw “the line of doom”
Put what you know on the “the line of doom” until
you have the units in what you want to know
Dimensional Analysis
• Straightforward Example
– How many pennies in 4 dollars?
• Know:
– 4 dollars
– 100 pennies = 1 dollar
• Want to know:
– Answer should be in pennies
100 pennies
1
1 dollar
4 dollar
4 dollar 
1
4 dollar
 1 = 4 dollar = ? pennies
1
4 dollar
100 pennies

 ? pennies
1
1 dollar
4 dollar
100 pennies 4×100 pennies 400 pennies



= 400 pennies
1
1 dollar
1 1
1
Dimensional Analysis
• Another Straightforward Example
– How many pennies in 4 dollars?
• Know:
– 400 pennies
– 100 pennies = 1 dollar
• Want to know:
– Answer should be in dollars
1 dollar
1
400 pennies
400 pennies
 1 = 400 pennies
400 pennies
1 dollar

100 pennies
400 pennies
1 dollar
400 dollar 4 dollar



= 4 dollars
100 pennies
100
1

Dimensional Analysis
• Yet Another Straightforward Example
– How many eggs in 2 dozen?
• Know:
– 2 dozen
– 12 eggs = 1 dozen
• Want to know
– Answer should be in eggs
12 eggs
=1
1 dozen
2 dozen
12 eggs


1 dozen
2 dozen
12 eggs 2×12 eggs


 24 eggs
1 dozen
1
Dimensional Analysis
• Lots of Examples
– How grams of flour do you need to make a loaf of
bread?
• 6.5 cups/loaf
• 1 cup flour = 125 g
– How many seconds in 1 year?
• 365.2425 days/year
– How long would it take me to walk home?
• 7.7 miles (Google Earth)
• 2.8 miles/hour
– Will I make it home by car?
• 0.1 of a tank of gas
• 18 mi/gal
• Gas tank holds 14 gal
Dimensional Analysis
• Useful Conversions
– Mass
•
•
•
•
1 lb = 453.6 g
1 g = 1000 mg
1 Kg = 1000 g
Kg = 2.205 lbs
– Volume
•
•
•
•
1 L = 1000 mL
1 cm3 = 1 mL
1 gal = 3.785 L or 1 L = 0.2642 gal
1 case of wine = 9.00 L = 2.378 gal
– Area
• 1 Hectare = 2.471 Acre
Dimensional Analysis
• Beyond Another Example
– How many pizzas do I need to order?
• 24 students + 1 instructor
• 1.5 slices/student or instructor
• 8 slices = 1 pizza
– About 75% of people will take a free sample of
dish soap at Costco. On average 4,250 people visit
Costco each day. How many samples should you
have per day?
Dimensional Analysis
• Beneath the Planet of Another Example
– How many 2L bottles of Diet Coke do you need to
make 1.6 gal?
– A solution has 4.200 mg/L of gold. How many
pounds per gallon?
Dimensional Analysis
• A Foreign Example
– You are in Barolo and you want to buy some pears
for lunch. Each pear weighs about 200g. They cost
€3.50 per Kilogram. Today the exchange rate is
€1.00 = $1.07(US). How much do the pears cost in
$(US)/lb?
Dimensional Analysis
• Wine Related Examples (finally!)
– How many gallons in a barrel of wine?
• Barrel of wine 225 L
– How many barrels will I need for my 2,500 gal ZN
– How many g of bentonite do I need to make 5.0 L
of a 6% weight to volume solution?
– How many cases of glass will in need for 2,500 gal
of wine?
• 9L/case
– How many bags of corks will I need for my 2,500
gal of ZN?
• 1000 corks/bag
Dimensional Analysis
• More Wine Related Examples
– How many Kg yeast to add to 4.2 tons of CS must?
• Mechanical harvest ~170 gal/ton
• Yeasting rate 2lb/1000g
– How many barrels can be filled in 6 hours?
• Fill one barrel in 8 minutes
– How much bentonite slurry do I add to my 3,478
gal of MA?
• 1.0 kg bentonite/8.0 L hot water slurry
• 600 mg/1.0 L dosage rate in the wine (ETS)
Dimensional Analysis
• Even More Wine Related Examples
– How much should you budget for 6 bottle boxes
for your Pinot noir that you will be bottling this
summer? You have 10 puncheons (300 L) and 6
barrels (228 L) of wine. 90% of the bottled wine
will be packed in six-pack lie down boxes. You
estimate that it will take 18 hours to do the
bottling. 1.5% of the wine will be lost in
evaporation, cellar operations, and bottling. A
bundle of 25 six pack lie down boxes costs $82.50.
Bottles are 750 ml. You hope to get 96 points from
The Wine Expectorator.
Next Week
• The Atom
• The Periodic Table