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Math and the Environment

Compute how conserving water can positively impact climate change.

Learning Objectives

After completing this section, you should be able to:

Compute how conserving water can positively impact climate change.
Discuss the history of solar energy.
Compute power needs for common devices in a home.
Explore advantages of solar power as it applies to home use.

Making a Positive Impact on Climate Change—Water Usage

Our use of water is one element that impacts climate change. Having access to clean, potable water is critical for not only our health but also for the health of our ecosystem. About 1 out of 10 people on our planet do not have easy access to clean water to drink. As each of us conserves water, we prolong the life span of fresh water from our lakes and rivers and also reduce the impact on sewer systems and drainage in our communities. Additionally, as we conserve water, we also conserve electricity that is used to bring water to and in our homes. So, what can we do to help conserve water?

Brushing Your Teeth (One Person’s Contribution)

Try it.

Brushing your teeth with the water running continually uses about 4 gal of water. Turning the faucet off when you are not rinsing uses less than one-fourth of a gallon of water. Considering the recommendation to brush your teeth twice a day, how much water would be saved in a week if the faucet was off when not rinsing?

Solution

Leaving the water running continually:

Step 1: Calculate gallons used not with water running continually:

Brushing twice a day for 7 days using 4 gal of water for each brushing

$(2\ \text{times a day})(7\ \text{days})(4\ \text{gal})=56\ \text{gal}$

Step 2: Calculate gallons used turning the faucet off when you are not rinsing:

Brushing twice a day for 7 days using 0.25 gal of water for each brushing

$(2\ \text{times a day})(7\ \text{days})(0.25\ \text{gal})=3.5\ \text{gal}$

Step 3: Calculate savings:

$\begin{array}{lll}\text{Savings} & = & 56\ \text{gal}-3.5\ \text{gal} \\ & = & 52.5\ \text{gal}\end{array}$

During one week, 52.5 gal of water would be saved if one person turned the faucet off except when rinsing when brushing your teeth.

Brushing Your Teeth (Multiple People’s Contribution – Town)

Try it.

Using the data in , how much water would be saved in a month if one-fifth of a town’s population of 15,000 turned the faucet off when brushing their teeth except when rinsing?

Solution

Step 1: From , we found that 1 person saves 52.5 gal per week.

Step 2: Calculate the population to save water:

$\text{One-fifth of}\ 15,000\ \text{people}=3,000\ \text{people}$

Step 3: One-fifth of a town’s population turning off the faucet when brushing their teeth for a month:

$(3,000)(52.5\ \text{gal per week})(4\ \text{weeks})=630,000\ \text{gal}$

During one month, 630,000 gallons of water would be saved if one-fifth of a town of 15,000 people turned the faucet off except when rinsing when brushing their teeth.

Condensed — the full section is in OpenStax Contemporary Mathematics.

History of Solar Energy

In the mid-1800s, Willoughby Smith discovered photoconductive responsiveness in selenium. Shortly thereafter, William Grylls Adams and Richard Evans Day discovery that selenium can produce electricity if exposed to the sun was a major breakthrough. Less than 10 years later, Charles Fritts invented the first solar cells using selenium. Jumping a mere 100 years later, Bell Labs in the United States produced the first practical photovoltaic cells in the mid-1950s and developed versions used to power satellites in the same decade.

Solar panel use has exploded in recent decades and is now used by residences, organizations, businesses, and government buildings such as the White House, space to power satellites, and various methods of transportation. One reason for the expansion is a continuing drop in cost combined with an increase in performance and durability. In the mid-1950s, the cost of a solar panel was around $300 per watt capability. Twenty years later, the cost was a third of the 1950s’ cost. Currently, solar panel cost has dropped to less than $1 per watt while decreasing in size as well as increasing in longevity. The dropping price and improved performance has moved solar to a modest investment that can pay for itself in less than half the time of systems from 15 years ago.

Solar Power’s Age

The sun has been harnessed by humans for centuries. The earliest recorded use of tapping the sun’s energy for power dates back to the seventh century BC when man focused the sun’s rays through a magnifying glass to create fire. Four thousand years later, we find historical record of using mirrors to focus the sun and light torches, often for ceremonial proceedings. Use of the sun to light torches continued through the centuries and has been recorded by various cultures including the Chinese civilization in 20 AD and beyond.

In more recent years, the sun was harnessed to power ovens on ships traversing to oceans in the 1700s. At the same time, the power of the sun was utilized to power steamboats through the 1800s. Mária Telkes, a Hungarian-born American scientist, invented a widely deployed solar seawater distiller used on World War II life rafts. Soon after, she partnered with architect Eleanor Raymond to design the first modern home to be completely heated by solar power. Air warmed on rooftop collectors transferred heat to salts, which stored the heat for later use.

Although solar panels as we know them today are relatively new in history, use of the sun to harness power is much older.

Compute Power Needs for Common Home Devices

A kilowatt (kW) is 1,000 watts (W). A kilowatt-hour (kWh) is a measurement of energy use, which is the amount of energy used by a 1,000-watt device to run for an hour. Using the definition of a kilowatt-hour, to calculate how long it would take to consume 1 kWh of power, we divide 1,000 by the watts use of a device.

For example, a 75 W bulb would take $1,000\div 75=13.3\ \text{hours}$ to use 1 kW of power.

Calculating the Kilowatt-Hours Needed to Run a Television

Try it.

A 48 in plasma television uses about 200 W. How many kilowatt-hours are needed to run the television in a month if the television is one for an average of 2.5 hours a day?

Solution

Step 1: $1,000/(200\ \text{watts})=5\ \text{hours}$ to use 1 kW

Step 2: $(2.5\ \text{hours a day})(30\ \text{days})=75\ \text{hours}$ of use

Step 3: $75\ \text{hour}/5\ \text{hours per kW}=15\ \text{kW}$

The television will consume about 15 kW in a month.

Calculating the Cost to Run a Refrigerator

Try it.

A medium-sized Energy Star–rated refrigerator uses about 575 W and runs for about 8 hours per day. What is the monthly (30 days) cost of running the refrigerator if the electric rate is 12 cents per kilowatt-hour?

Solution

Step 1: Calculate the watts per day:

$(575\ \text{W})(8\ \text{hours})=4,600\ \text{W per day}$

Step 2: Calculate the kilowatt-hours.

$(4,600)/(1,000)=4.6\ \text{kWh}$

Step 3: Calculate the daily cost.

$(4.6\ \text{kWh})(12\ \text{cents})=55\ \text{cents}=\text{\$}0.55$

Step 4: Calculate the monthly cost.

$(\text{\$}0.55)(30)=\text{\$}16.50$

It would cost about $16.50 to run the refrigerator for a month.

Calculating the Kilowatt-Hours to Run an Oven

Try it.

An electric oven is labeled as 4,000 W. How much would it cost to bake a cake for 30 minutes if the electric rate is 14 cents per kilowatt-hour?

Solution

Step 1: Determine the time it takes to use 1 kW of power:

$1,000/(4,000\ \text{watts})=0.25\ \text{hours to use}\ 1\ \text{kW}$

For every 15 minutes, the oven uses 1 kW of power.

Step 2: Determine how many kilowatt-hours are needed to bake the cake for 30 minutes:

$(30\ \text{minutes})/(15\ \text{minutes per kW})=2\ \text{kW}$

Step 3: Calculate the cost of the oven usage:

$(2\ \text{kW})(14\ \text{cents per kWh})=28\ \text{cents}$

It would cost about 28 cents to bake the cake.

Solar Advantages

There are multiple advantages that solar power can offer us today including reducing greenhouse gas and CO₂ emissions, powering vehicles, reducing water pollution, reducing strain on limited supply of other power options such as fossil fuels. We will look further at reducing greenhouse gas and CO₂ emissions.

Any gas that prevents infrared radiation from escaping Earth's atmosphere is a greenhouse gas. There are 24 currently identified greenhouse gases of which carbon dioxide is one. When measuring the impact of any of the greenhouse gases, the measurements are given in units of carbon dioxide emissions. For this reason, greenhouse gas and carbon dioxide have become interchangeable in discussions.

Charles Fritts and Mohammad M. Atalla

Charles Fritts, a New York inventor, is credited with creating the first solar cell, which he installed on a rooftop in New York City in 1884. While the solar cell was not very efficient, having a rate of conversion between 1 to 2%, this was a major step early in solar power energy. Today’s solar cells have an efficiency on average of 15 to 20%, which yields a notably higher impact. Nonetheless, the work that Fritts successfully completed marked the start of solar energy through the use of photovoltaic solar panels in the United States.

Mohamed M. Atalla was an Egyptian-born scientist who moved to the United States to complete his studies, and undertook research and development at Bell Laboratories in New Jersey. Many of the early efficiency gains in solar cells were due to his development of processes for using silicon within electronic devices. Atalla's work led to the invention of silicon transistors and microchips (including his own invention of the MOSFET, the most widely used transistor in the world), and quickly increased the efficiency of solar cells.

Calculating the Solar Power for Average Home Use in Kilowatts

Try it.

If a home uses approximately 30 kW of electricity per day, what size solar system would be needed to fuel 80% of a home’s needs for a month (30 days)?

Solution

$(30\ \text{kW hours})(30\ \text{days})(0.80)=720\ \text{kW}$

A solar system capable of producing 720 kW a month would be needed.

Key Concepts

Recognize how water conservation by one person, family, community, or nation can positively impact the world’s freshwater supply.
Recall components from the history of solar use by mankind.
Calculate electrical demand given watts.
Recognize advantages of residential solar power.

Practice (7)

Try each one on paper first. Reveal the answer to check; verified ones can be opened in the solver for every step.

Brushing your teeth with the water running continually uses about 4 gal of water. Turning the faucet off when you are not rinsing uses less than one-fourth of a gallon of water. Considering the recommendation to brush your teeth twice a day, how much water would be saved in a week if the faucet was off when not rinsing?

Afslør svaret

Leaving the water running continually:
Step 1: Calculate gallons used not with water running continually:
Brushing twice a day for 7 days using 4 gal of water for each brushing
$(2\ \text{times a day})(7\ \text{days})(4\ \text{gal})=56\ \text{gal}$
Step 2: Calculate gallons used turning the faucet off when you are not rinsing:
Brushing twice a day for 7 days using 0.25 gal of water for each brushing
$(2\ \text{times a day})(7\ \text{days})(0.25\ \text{gal})=3.5\ \text{gal}$
Step 3: Calculate savings:
$\begin{array}{lll}\text{Savings} & = & 56\ \text{gal}-3.5\ \text{gal} \\ & = & 52.5\ \text{gal}\end{array}$
During one week, 52.5 gal of water would be saved if one person turned the faucet off except when rinsing when brushing your teeth.
Using the data in , how much water would be saved in a month if one-fifth of a town’s population of 15,000 turned the faucet off when brushing their teeth except when rinsing?

Afslør svaret

Step 1: From , we found that 1 person saves 52.5 gal per week.
Step 2: Calculate the population to save water:
$\text{One-fifth of}\ 15,000\ \text{people}=3,000\ \text{people}$
Step 3: One-fifth of a town’s population turning off the faucet when brushing their teeth for a month:
$(3,000)(52.5\ \text{gal per week})(4\ \text{weeks})=630,000\ \text{gal}$
During one month, 630,000 gallons of water would be saved if one-fifth of a town of 15,000 people turned the faucet off except when rinsing when brushing their teeth.
Using the data in , how much water would be saved in a year if one-fourth of the population of the state of Minnesota, which is approximately 5.6 million people, turned the faucet off when brushing their teeth except when rinsing for a year (52 weeks)?

Afslør svaret

Step 1: From , we found that 1 person saves 52.5 gal per week.
Step 2: Calculate the population to save water:
$\text{One-fourth of}\ 5.6\ \text{million people}=1.4\ \text{million people}$
Step 3: One-fourth of a town’s population turning off the faucet when brushing their teeth for a month:
$(1.4\ \text{million})(52.5\ \text{gal per week})(52\ \text{weeks})=3,822\ \text{million gal}$
During one year, 3,822 million gal of water would be saved if one-fourth of the state of Minnesota turned the faucet off except when rinsing when brushing their teeth.
A 48 in plasma television uses about 200 W. How many kilowatt-hours are needed to run the television in a month if the television is one for an average of 2.5 hours a day?

Afslør svaret

Step 1: $1,000/(200\ \text{watts})=5\ \text{hours}$ to use 1 kW
Step 2: $(2.5\ \text{hours a day})(30\ \text{days})=75\ \text{hours}$ of use
Step 3: $75\ \text{hour}/5\ \text{hours per kW}=15\ \text{kW}$
The television will consume about 15 kW in a month.
A medium-sized Energy Star–rated refrigerator uses about 575 W and runs for about 8 hours per day. What is the monthly (30 days) cost of running the refrigerator if the electric rate is 12 cents per kilowatt-hour?

Afslør svaret

Step 1: Calculate the watts per day:
$(575\ \text{W})(8\ \text{hours})=4,600\ \text{W per day}$
Step 2: Calculate the kilowatt-hours.
$(4,600)/(1,000)=4.6\ \text{kWh}$
Step 3: Calculate the daily cost.
$(4.6\ \text{kWh})(12\ \text{cents})=55\ \text{cents}=\text{\$}0.55$
Step 4: Calculate the monthly cost.
$(\text{\$}0.55)(30)=\text{\$}16.50$
It would cost about $16.50 to run the refrigerator for a month.
An electric oven is labeled as 4,000 W. How much would it cost to bake a cake for 30 minutes if the electric rate is 14 cents per kilowatt-hour?

Afslør svaret

Step 1: Determine the time it takes to use 1 kW of power:
$1,000/(4,000\ \text{watts})=0.25\ \text{hours to use}\ 1\ \text{kW}$
For every 15 minutes, the oven uses 1 kW of power.
Step 2: Determine how many kilowatt-hours are needed to bake the cake for 30 minutes:
$(30\ \text{minutes})/(15\ \text{minutes per kW})=2\ \text{kW}$
Step 3: Calculate the cost of the oven usage:
$(2\ \text{kW})(14\ \text{cents per kWh})=28\ \text{cents}$
It would cost about 28 cents to bake the cake.
If a home uses approximately 30 kW of electricity per day, what size solar system would be needed to fuel 80% of a home’s needs for a month (30 days)?

Afslør svaret

$(30\ \text{kW hours})(30\ \text{days})(0.80)=720\ \text{kW}$
A solar system capable of producing 720 kW a month would be needed.

Symbols used here

factorial
n × (n−1) × … × 1; the number of orderings of n things. 0! = 1.

\binom{n}{k}

binomial coefficient, "n choose k"
Number of k-element subsets of n things: n!/(k!(n−k)!).

\sum_{k=1}^{n} a_k

summation
Add a_k for k = 1 up to n.

x \in A,\ A \subseteq B

element of, subset
x belongs to A; every element of A is in B.

A \cup B,\ A \cap B,\ A \setminus B

union, intersection, difference
In either; in both; in A but not B.

\emptyset,\ |A|

empty set, cardinality
The set with no elements; the number of elements of A.

\forall,\ \exists

for all, there exists
Quantifiers: every x; at least one x.

\neg,\ \wedge,\ \vee,\ \Rightarrow,\ \Leftrightarrow

not, and, or, implies, iff
Logical connectives.

\blacksquare\ \text{or}\ \square

end of proof (halmos)
Marks the point where the statement has been established.

a \equiv b \pmod n

congruent modulo n
n divides a − b; a and b have the same remainder.

O(n^2),\ \Theta,\ \Omega

big-O notation
Grows no faster than n² (up to a constant), for large n.

a \bmod n

remainder
What is left after dividing a by n.

How to: Math and the Environment

Compute how conserving water can positively impact climate change.
Discuss the history of solar energy.
Compute power needs for common devices in a home.
Explore advantages of solar power as it applies to home use.
greenhouse gas
CO
watt
kilowatt (kW)

Questions people ask

What makes mathematics "discrete"?

It deals with separate, countable objects — integers, graphs, statements — rather than continuous quantities. No limits, no infinitesimals; instead induction, counting and logic.

How does a proof by induction work?

Show the statement for the first case, then show that whenever it holds for n it holds for n + 1. Like dominoes: the first falls, and each knocks over the next.

Prøv din egen

Parts of this page are adapted from OpenStax Contemporary Mathematics (CC BY-NC-SA 4.0). Condensed and re-explained here; errors are ours.

Mere i Discrete Math & Logic

Truth tables Sums and induction Proof by induction Algorithms and growth of functions