Chapter 3 The Algebra of Sustainability
Answers to selected problems
Dimensional Analysis
Name _____________________________ Points _____/16 Attendance ___/2 Total ___/18
Note: Use the exact equivalencies from the table and be precise in showing the process.
In problems 1, 2 and 3, the entire dimensional analysis problem has been provided; you
only need to calculate the answer. Multiply by numbers in the numerator and divide by numbers
in the denominator to obtain the converted value. Ignore the ones.
(1)
1. Lengths: In a normal home, the ceilings of a room are 8 feet above the floor. What
is this distance in meters?
 12in   2.54cm  1m 
8 ft 


  _ 2.4384 _ meters hint (8·12·2.54/100)
 1 ft   1in  100cm 
(1)
2. Area: A home contains 2000 square feet. How many square meters is the house?
Remember to square the unit fractions.
2
 12in   2.54cm   1m 
2
2
2
2
2000 ft 
 
 
  _ 185.8 _ m hint(2000·12 ·2.54 /100 )
 1 ft   1in   100cm 
(1)
3. Mixed: A person consumes approximately 2000 kilocalories per day. How many
kilocalories are required by a community of 500 people for a year?
 2000kcal   500 people   365days 
 kcal 


  _ 365, 000, 000 __ 


  1year 
 person  day  
 year 
2
2
2
In problems 4 and 5, use the Unit Equivalencies table to put in the numbers missing from
the unit fraction, then cancel units that are the same in the numerator and denominator and
multiply or divide the numbers, as appropriate.
(2)
4. Energy: Household energy consumption is calculated by multiplying the amount of
energy needed times the number of hours that it is used. Your electric bill is calculated based on
the number of kilowatt hours (kWh) that you use. The energy requirements of most appliances
are measured in watts, while the time they are used is often measured in minutes, thus it is
necessary to convert from watt minutes to kilowatt hours. If you know the cost of energy, you
can determine how much it costs to operate an appliance.
A 1250 watt microwave oven uses 1250 watts of energy. If it is turned on for 24 minutes during
the course of a day, how much energy was used in units of kilowatt hours?
 _ 1 ___ kW  __ 1 __ h 
1250W  24 min 

  _ 0.5 _ kWh
 _ 1000W   _ 60 __ min 
Math In A Sustainable Society 2.0
(2)
5. Energy Costs: If a clothes dryer uses 4.5 kilowatts of energy and the dryer is
operated for 1 hour and 10 minutes how much does it cost to dry the clothes if the cost of energy
is $0.06 per kilowatt hour.
 ____ hr   $ _____ 
4.5kW  70 min 

  $ __________
 ____ min   kWh 
In problems 6 to 8, do the entire problem yourself using dimensional analysis. In all cases, show
the appropriate dimensional analysis procedure as demonstrated above and then complete the
multiplication.
(3)
this?
6. Volume: A person is supposed to drink 64 ounces of water a day. How many liters is
(3)
7. Volume: A home contains 500 cubic meters of space. What is the volume in cubic
feet? Remember to cube the unit fractions.
(3)
8. Mixed: If a piece of land used for farming can produce 4000 kilocalories of energy
per day per acre, then how many kilojoules of energy does it produce per day per hectare?
Math In A Sustainable Society 2.0
Determine the shape of homes in Steilacoom Valley
Name________________________Points _____/16 Attendance ___/3 Total ___/19
Homes in the United States range in size from under 100 m2 to over 400 m2. Given that
there was a time when people lived in much smaller houses, one may wonder what the smallest
sized home is that can be comfortable for a family.
(1) 1. Use dimensional analysis to determine the number of square feet in a home that has an
area of 100 square meters. The equivalencies you need are 100 cm = 1 m, 2.54 cm = 1in, 12 in =
1 ft. Be careful, because of the squared units in this problem.
Assumptions for Steilacoom Valley homes:
1. Families will contain 3 or fewer people (one child at most because we are trying to limit the
population).
2. In a sustainable community, some items will be shared, so each house doesn’t need one.
Also, the amount of “stuff” a person has can be minimized.
3. Every family wants their own house, but to use less land, houses will be close to each other.
4. A home that contains the most area inside with the least amount of outer wall space is the
ideal for sustainability because less material is used to build the walls and less heat is lost
through the walls if there is less wall space.
Shape: What is the best shape for a home? Let’s experiment with a home that has an area of 36
square meters. The formulas we will use are A=LW, P = 2L + 2W, A = πr2, C = 2πr. Determine
the perimeter for each of the following shapes:
(1)
(1)
(1)
(1)
(1)
Rectangle: 1 x 36
Rectangle: 2 x 18
Rectangle: 3 x 12
Rectangle: 4 x 9
Square: 6 x 6
(1) Circle: r=3.385
Perimeter = 74
Perimeter = 40
Perimeter = 30
Perimeter = 26
Perimeter = 24
Circumference = 21.27
(1) 2. What do you conclude is the best shape of a house for maximizing the area while
minimizing the distance around (perimeter or circumference)?
Circle 1: Rectangle
Square
Circle
Now consider that a house is not two dimensional (length and width) but it is three
dimensional (length, width and height). Therefore, the walls and roof, all of which require
material to build and all of which are sources of escaping energy, must be considered.
Furthermore, the volume of air inside must also be considered as larger air volumes require
greater amounts of heat. Since a square was the best rectangular shaped area, lets compare a
Math In A Sustainable Society 2.0
square home with 8 foot high walls and a flat roof to a round home built like a dome. This is like
half a sphere.
(1) 3. Convert the height of 8 feet to meters.
2.43 m
(1) 4. What is the volume of a 6 meter x 6 meter home that has an 8 foot ceiling? 87.78 m3
(1) 5. What is the area of the outer walls and roof? 94.52 m2
If a dome (half sphere) is used with the radius of 3.385 m, then the volume of the dome can be
14

calculated using the formula V    r 3  . The area of the outer walls can be computed using
23

1
A   4 r 2 
2
(1) 6. What is the volume of the dome?
81.23 m3
(1) 7. What is the area of the outer walls? 71.99 m2
8. Given that both the square and dome home have the exact same floor area, answer the
following questions to determine the better design.
(1) 8a. Which has less air to heat inside?
Square
Dome
(1) 8b. Which has less wall area through which heat is lost?
Square
Dome
(1) 8c. Which is more sustainable?
Square
Dome
This dome, which was designed by Cloud Hidden
Designs, LLC, was the winner of the Domes for the World
Design Challenge in 2008. Design parameters required
that the diameter of the single family homes must be less
than 40 feet. The cost must be less than $2,500. The
objective was to provide affordable housing in areas of the
world that suffer from poverty and natural disasters that
destroy the local homes.
Math In A Sustainable Society 2.0
Determining the Carrying Capacity of Steilacoom Valley
Part 1. Finding a function for the amount of land for homes per resident
Name____________________________Points _____/6 Attendance ___/1 Total ___/7
Show work for all problems.
Public buildings
To be a vibrant community, some public facilities are necessary. These might include a
library, community center, activity room/theater, laboratory and medical facility.
It would be very difficult to design public buildings as we don’t have enough
information, so we will just use an estimate of 20,000 m2, which is the size of the Tacoma Dome.
This should provide enough space for all public activities.
We will also estimate that workshops such as the butcher shop, bakery, furniture and
cabinetry shops, etc will occupy a facility of about 10,000 m2.
Carrying Capacity
The carrying capacity of Steilacoom Valley will be determined based on the following
assumptions.
1. Only 20% of all the land in Steilacoom Valley will be developed. The remaining land may be
used for hiking, snowshoeing or similar activities but in general it will be the amount of land that
will not be developed in anyway, ever. It will be the land that residents “allow” nature to have.
2. Food will be grown to meet the annual needs of the community, but not for export or long
term storage.
3. Land will be needed for housing, public activities, and shops
4. A safety factor of 50% will be included in the amount of land needed to account for space
between buildings and other additional area.
5. There will not be any motor vehicles or roads.
The total land area needed for SV residents is given by:
Land =1.5 (Housing + Public Building + Shops + Farmland)
This can be expressed as the combination of functions for which H(R) is a function for the land
needed for housing and F(R) is a function for the amount of farmland needed based on the
number of residents.
L(R) = 1.5 (H(R) + P + S + F(R))
Math In A Sustainable Society 2.0
Determine the land area needed for the homes, as a function of the number of residents.
Assumption 1: Each home will house an average of 2 people.
Assumption 2: All homes have an area of 50 m2.
(1) 1. Determine the radius of a round home with an area of 50 m2. Round your answer to the
nearest whole number.
R=4
(1) 2. If we plan to build homes close to each other by placing them
inside a square property with the side length equal to the diameter of the
house plus 2 meters, so that houses are about 2 meters apart, then how
much land would each home require?
A = 100 m2
(1)
3a. If there are 1000 residents, how much area will be required for the homes based on
our assumption of an average of 2 people per house?
A = 50,000 m2
(1)
3b. If there are 1200 residents, how much area will be required for the homes based on
our assumption of an average of 2 people per house?
A = 60,000 m2
(2)
3c. Generalize this by writing the function H(R) to show how much area will be required
for the homes as a function of the number of residents for any number R. H(R) should have units
of square meters. Simplify completely.
H(R) = 50R
Record your answer to 3c on the top of the Part 2 before turning in this activity.
If you complete this page during class, begin Part 2 of this Carrying Capacity Activity.
Math In A Sustainable Society 2.0
Determining the Carrying Capacity of Steilacoom Valley
Part 2. Finding a function for the amount of farmland and finding the carrying capacity.
Name____________________________Points _____/15 Attendance ___/2 Total ___/17
Show All Work
From Part 1: What is the function H(R) = 50R
Determine the amount of farmland needed for Food Production
Assumption 1: The average person in Steilacoom Valley will consume 2,500 kilocalories per
day.
Assumption 2: Grain and vegetables will be grown on a piece of land only once every four
years. During the three years it isn’t being used, it will be allowed to grow over (fallow) and can
be used by grazing animals such as bison, sheep, lamas, goats, and poultry. These animals will
be used for meat and wool, milk, eggs etc. Allowing the animals on this unused (and rotated)
farmland will result in natural fertilization. Most commercial fertilizer is produced from natural
gas, which we are assuming is no longer available.
6000kcal
Assumption 3: An estimated 6000 kilocalories can be produced per day per acre
day  acre
when growing grains and vegetables. The estimate for meat is about 1200 kilocalories per day
1200kcal
per acre,
.
day  acre
4.
To determine the total amount of farmland needed per person solve the two simultaneous
equations.
Equation 1: Total Calories per person per day = Vegetable/grain Calories + Meat Calories
2500 = 6000V + 1200M
where
V = number of acres for Vegetables/grain
M = number of acres for Meat
Equation 2: M = 3V since the number of acres for meat = 3 times the amount of land for
vegetables/grains.
(2)
4a. How many acres are needed per person for vegetables and grain?
0.26 acres
(1)
4b. The total number of acres needed per person is given by N = 4V.
1.04 acres
(2)
4c. Convert the number of acres per person to square meters per person. Round to the
nearest whole number.
4209 or (4215.6) depending on rounding of prior answers.
Math In A Sustainable Society 2.0
(2)
4c. Generalize this by writing a function F(R) for which the amount of farmland needed
per person is a function of the number of residents. The units should be square meters.
F(R) = 4209R or F(R) = 4215.6R
(2)
5a. Simplify our land requirement function L(R) = 1.5 [H(R) + P + S + F(R)] by using
the generalized results from 3 and 4 to replace H(R) and F(R). Replace P and S with their
values. Combine all like terms and distribute the 1.5. Write the most simplified form of L(R).
L(R) = 6398.4R + 45,000
or
L(R) = 6388.5R + 45,000
The units for L(R) are square meters. This function represents the land requirement for each
resident, under the assumptions that have been made.
(2)
5b. Since land area is usually expressed in hectares, then rewrite the function by
converting the numbers to hectares. 1 hectare = 10,000 square meters
L(R) = 0.63984R + 4.5
or
L(R) = 0.63885R + 4.5
(2)
6. The total land area of Steilacoom Valley is 80 square kilometers. Only 20% of the
land will be developed for human use. What is the largest amount of land, in hectares, that could
be developed by the settlers? 100 hectares = 1 square kilometer.
1600 hectares
(2)
7. Use the Land function for the amount of land you found in problem 6 to determine the
carrying capacity, by solving for R.
R = 2494 residents or 2498 residents.
Math In A Sustainable Society 2.0
The Algebra of Sustainability
Energy
Name____________________________Points _____/15 Attendance ___/2 Total ___/17
At the most basic level, survival of all living organisms is dependent upon a regular
influx of energy. Most living organisms get this energy from the food they eat. It is measured in
calories. Of all the species, only one has been able to create enhanced living conditions by using
the earth’s storehouse of energy. This storehouse contains petroleum, coal, natural gas and
uranium. By using the energy stored in these resources, humans have been able to create a world
where many people can do more than simply survive. This storehouse of energy contained only
a few hundred year’s supply, given the size of the world population. It allows us to refrigerate
and cook food, heat our homes and water, wash clothes, use machinery and electronic products,
etc. The entire motivation for the Steilacoom Valley project is to relearn how to live in a world
without this stored energy, which at some point, will be completely consumed. Energy, in many
ways, is the key to life.
Determine the energy requirements for Steilacoom Valley.
Energy is measured in units of kilowatt hours. These units are multiplied and reflect the amount
of energy being consumed over a period of time. All the energy used in Steilacoom Valley will
be generated using windmills.
Home Energy Use
We will make certain assumptions when determining energy requirements.
 Homes are small and well insulated so they only contain a small heating element.
 Food is prepared in communal eating areas, not individual kitchens, but they do contain
individual small cooking appliances (burner, toaster oven).
 Laundry is washed in designated areas so everyone does not need a washer/dryer
6 light bulbs
1 tankless water heater
Space Heater
Cooking appliances
Estimated daily consumption
13 watts per bulb, 4 hours
10 kWh/day
1500 watts, 1 hours
1000 watts, 0.5 hours
1. Determine the daily energy use per house in kWh. (Show work, be organized)
(1) Light bulbs
Water heater
10 kWh
(1) Space Heater
(1) Cooking appliances
(1) Total
Math In A Sustainable Society 2.0
(2)
2. Determine the community’s daily home energy use, assuming 500 houses.
(2)
3. If the public buildings use 10,800 kWh per day (this is approximately the amount used
by the Pierce College Puyallup Campus), then how much energy is used by Steilacoom Valley?
Assume all the electrical energy will be produced by windmills. Also, assume the
average wind speed is 18 mph. The turbines will produce a maximum power of 1000 KW of
energy with a 54 meter blade span. http://science.howstuffworks.com/wind-power4.htm
http://www.awea.org/faq/basicpp.html
(2)
4. Use the Power Curve to estimate the actual turbine output for an average wind speed
of 18 mph. Show this on the graph. Change the percent to a proportion then multiply times
1000 kW.
(1)
5. Multiply the turbine output times 24 hours to determine the average number of kWh
produced by each windmill in a day.
(2)
up.
6. How many of these windmills will be needed to meet the community needs? Round
(2)
7. Use a safety factor of 50% to determine how many windmills should be built. This
will allow for shut down due to problems or maintenance.
Math In A Sustainable Society 2.0
Chapter 3 Homework
Name __________________________________ Points ____/___
1. Determine the carrying capacity of Steilacoom Valley using the following changes of
assumptions. All other assumptions in the carrying capacity activity will remain unchanged.
Change the average home size from 50 m2 to 70m2.
Change the average number of people per house to 3.
Change the calories per acre for grain/vegetables to 4,500 kcal/(day·acre)
Change the kilocalories per acre for meat to 1000 kcal/( day·acre)
Change the crop rotation to every 3 years, thus there are two fields for meat and one for
grain/vegetables.
Math In A Sustainable Society 2.0
2. Determine the number of windmills needed for 2500 people if there is an average of 2.5
people in each home. All other assumptions in the energy activity will remain unchanged.
Math In A Sustainable Society 2.0
3.
Modern society has presented us with a paradox. A century ago, if you wanted to go
somewhere relatively close to home, you either walked or used a horse. Consequently most
people were relatively fit. Since then, the invention of the automobile has allowed us to travel
greater distances in less time. We have become used to using it for even short distances.
Walking to a destination became a strange concept for many (why are you walking, don’t you
want to drive?). Of course driving, along with some of the other sedentary things we do, has led
to lower levels of fitness. Our solution for that is to drive to a favorite fitness center then
exercise on a treadmill or ride a stationary bike. While it might take longer to walk or bike to
our destination, we might be able to save a lot of time (not to mention money and resources) by
not needing to go to the gym. This problem can be used to determine which approach saves the
most time. To do so, we need to start with some basic assumptions.
Assumptions:
Assume that work is a distance of 15 miles and that because of lights, the type of roads
and congestion, the average speed is 30 miles per hour. Assume that the distance to the fitness
center is 12 miles from your home, that your average speed is also 30 miles per hour and that you
exercise for 1 hour and take an extra half hour for changing and showering. The time to work
equals the time from work. The time to the fitness center equals the time from the fitness center.
We will then compare this to bicycling to and from work every day and not using the fitness
center at all.
Organize your thoughts:
To help organize our thinking, keep in mind the things we spend time doing. We spend
time going to work, coming home from work, going to the fitness center, exercising, etc, and
returning from the fitness center. We will determine the total time involved if we use a car and if
we use a bike. You will need the formula d=rt (distance = rate·time).
Total Time = Time to work + time from work + time to fitness Center + time at fitness Center +
time to home.
Let T = Total Time
W1 = Time to work
W2 = Time from work
F1 = Time to fitness Center
F = Time at fitness Center
F2 = Time from fitness Center
T = W1 + W2 + F1 + F + F2
Math In A Sustainable Society 2.0
Calculations (show work):
a. Find the time to work using a car, W1.
t = 0.5 hours
b. Find the time from work using a car, W2
t = 0.5 hours
c. Find the time to the fitness center, F1.
d. Find the time from the fitness center, F2.
e. Find the Total time, T when using a car.
T = 3.3 hours or 3 hours and 18 minutes
If we chose to bicycle to work, allowed 30 minutes for a shower after we got there and bicycled
home, we would get our cardio workout and not have to use the fitness center at all, but how
much time would be involved? Assume we can bicycle at 12 miles per hour.
f. Find the time it takes to bicycle to work W1. Include a half hour to shower at work.
g. Find the time it takes to bicycle home from work W2.
t = 1.25 h
h. Find the Total time, T when using a bicycle.
i. Based on the assumptions presented in this problem, will driving or bicycling take the least
time?
Math In A Sustainable Society 2.0
4. One of the hallmarks of suburbia is the grass yard. According to Adele Weder, writer for The
Tyee, an independent daily online magazine for British Columbia, a yard is “a kind of feudal
crest, marking the ability to own extravagantly useless land”. For many, this part the property
demands resources such as fertilizer, weed killers and water and after using these to help the
grass grow it then requires the homeowner’s sweat and time for maintenance along with gasoline
for mowing. As the price of gas and food climbs, home owners may reconsider the importance
of a perfectly manicured lawn.
Suppose a lawn mowing service, aware that business is decreasing, decides to expand
their services by providing a service in which homeowners convert their grass yard into
productive land using permaculture. The Permaculture Institute (www.permaculture.org) defines
permaculture as “… an ecological design system for sustainability in all aspects of human
endeavor. It teaches us how build natural homes, grow our own food, restore diminished
landscapes and ecosystems, catch rainwater, build communities and much more.”
In Washington there is minimal rainfall in July, August and September. One of the new
services of the lawn mowing company is to calculate the anticipated water needs for the gardens
they install. They then design a system to collect and store water that lands on the roof of the
house during the rainy months. This is an alternative to letting the water run off to the streams
and the Sound and reduces the demand on city water. The objective is to determine the water
needs and the amount of rain that will be needed to store enough water.
Below is a diagram of the property of one of the company’s clients.
Shadow Area
N
Radius of home: 18 ft
The property measures 80 x 136 feet. The round home has a radius of 18 feet and the shadow
area is a trapezoid. The long side of the trapezoid is 80 feet, the short side is 36 feet and the
height is 40 feet. The house has a radius of 18 feet.
Math In A Sustainable Society 2.0
Assumptions: All of the property except for the house and the shadow area will be planted and
will need watering. The area that is watered will need to receive one inch of water, twice a week
for 10 weeks.
Organize your thoughts:
 Find the area that must be watered by finding the area of the property, then subtracting
the shadow area and half the area of the home (there is an overlap of the shadow
trapezoid and half the home).
 Find the number of inches of water that must be applied.
 Use the number of inches of water and the area of the home to find the number of cubic
feet of water needed.
 This is the amount of water that must be stored.
 How much rain must fall on the roof during the rest of the year to collect enough?
 Divide the volume of water needed by the area of the roof. Convert your answer to
inches.
Calculations (show work)
a. Find the area of the property.
A = 10,800 ft2
( B  b) h 

b. Find the area of the trapezoid shadow  A 
.
2


c. Find the area of the home.
d. Find the area of the property that will need to be watered.
A = 8051 ft2
e. Use dimensional analysis to find the number of feet of water that must be applied during the
10 week period.
f. Find the volume of water that must be applied during the 10 weeks.
V = 13,418 ft3
g. Find the number of inches of rain that must fall on the house to accumulate enough water.
h. If the house is located in a place that receives an average of 35 inches of rain a year, will the
owner be able to store enough water from roof runoff?
Math In A Sustainable Society 2.0
5. In the Presidential Debate that occurred on October 15, 2008, both candidates answered the
following question asked by moderator Bob Schieffer. Would each of you give us a number, a
specific number of how much you believe we can reduce our foreign oil imports during your first
term? McCain’s answered “So I think we can easily, within seven, eight, ten years, if we put
our minds to it, we can eliminate our dependence on the places in the world that harm our
national security if we don't achieve our independence.” Obama’s answer was “I think that in
ten years, we can reduce our dependence so that we no longer have to import oil from the Middle
East or Venezuela. I think that's about a realistic timeframe.”
To gain some appreciation for what this would mean, let’s modify the question slightly
and determine the impact of reducing oil consumption in the US to a level in which we will not
have to import any oil. That is, we will only use the oil pumped from wells in the United States.
This is not an unreasonable assumption as there will come a time when foreign countries will
want to conserve their oil resources for their own country to use and so won’t sell them to the
US. We will use the timeframe of 10 years as stated by President Obama.
The graph below shows historical US petroleum production and consumption data. It is
based on data from the Energy Information Administration website.
U.S. Crude Oil Daily Production and Consumption
24
20
18
16
14
12
10
8
6
4
M o n th - Y e a r
U.S. Field Production of Crude Oil (Million Barrels Per Day)
U.S. Field Consumption of Crude Oil (Million Barrels Per Day)
Math In A Sustainable Society 2.0
Jan 2010
Jan 2000
Jan 1990
Jan 1980
Jan 1970
Jan 1960
Jan 1950
Jan 1940
0
Jan 1930
2
Jan 1920
Quantity (Million Barrels Per Day)
22
a. To model this problem, we will need a linear equation. In January 1990, the US produced 7.5
million barrels of Oil per day. In January 2000, the US was only able to produce 5.8 million
barrels of oil per day. Find the equation of the line through these two points then use the
equation to predict the amount produced in 2019 (10 years after the Obama Presidency began).
y = -0.17 x + 345.8
b. In January, 2009, the US consumed 19.1 million barrels of oil a day. To be completely off of
foreign oil in 10 years, we would have to reduce our consumption to the level you calculated in
the prior question. Find the slope of the line connecting the point (2009,19.1) to the production
amount you found for 2019.
The slope of the line you just calculated is the number of million of gallons of petroleum used
per day that we would have to reduce as a nation.
c. What is percent reduction would occur in the first year?
8.65%
d. For every thousand cars on the road in 2009, how many could not be used in 2010?
e. For every hundred days you drive in 2009, approximately how many of those days could you
drive in 2010?
91.3
f. What is the percent reduction that would occur after 10 years?
g. For every thousand cars on the road in 2009, how many could not be used in 2019?
865
h. How would you survive with this type of a reduction?
Math In A Sustainable Society 2.0