MATH For Environmental Science

Dimensional Analysis

Dimensional Analysis (Factor-Label)It is critical for success on the exam that you have a thorough understanding of dimensional analysis.

Height Activity

Use Dimensional Analysis or factor/label method for calculations

The following formula based on the cancellation of units is useful:

Given Value x Conversion factor =Answer 1

ORold unit x new unit = new unit

1 old unitExample:

25 ft x 1 yd x 1.094 m = 9.117 meters 3 ft 1 yd

Sample Question

Heating a house in Virginia. Assumptions:

• 50,000 BTUs of heat per square foot are required to heat the house for the winter.

• One cubic foot of natural gas supplies 1,000 BTUs of heat energy.

• Natural gas is available at a cost of $4.00 per thousand cubic feet.

• The house has 3,000 square feet of living space.

Sample Question for Dimensional Analysis

i. The number of cubic feet of natural gas required to heat the house for one winter

1. This problem wants to know cubic feet of natural gas needed each winter to heat the house. In math terms, it looks like this:

Sample Question for Dimensional Analysis

2. We know the house is 3,000 square ft. We also know that it takes 50,000 BTUs of heat per square foot to heat the home. Finally, we know that one cubic foot of natural gas supplies 1,000 BTUs of heat.

or and

  or and

 

Sample Question for Dimensional Analysis

3. The next step is to pick the statements that will cancel the units you do not want. You want ft3 in your answer. You need to eliminate BTUs and ft2.

or and

  or and

 

Sample Question for Dimensional Analysis

4. Now you need to eliminate BTUs. To do this you will need to use the equivalency that has BTUs on the bottom and cubic feet on the top.

=

natural gas

Sample Question for Dimensional Analysis

5. For part ii, you need an additional statement.

or  

Energy Efficiency

Systems will never be 100 % efficient (unless stated), so pay careful attention.

or

Energy EfficiencyNow, if we add a third part to the question stating the system is 50 % efficient, then the output must be divided by the efficiency to determine how much input energy is needed. REMEMBER: A percentage efficiency must be changed to its decimal equivalent.

natural gas

Scientific Notation

A number written in scientific notation consists of a coefficient and an exponent. Coefficients need to be between 1 and 9. The coefficient is then multiplied by ten raised to an exponent, 10exponent.

Determine the exponent on the “10” by counting the number of places you move the decimal point. If you move the decimal to the right, the exponent will be negative. If you move the decimal point to the left, the exponent will be positive.

Converting to Scientific Notation

575,000 = 5.75 × 105

1

Adding and Subtracting in Scientific Notation

First, make sure both numbers have the same exponent. Next, add the coefficients.

2.4 × 104 + 4.0 × 105=

.24 × 105 (moving the decimal to the left made the exponents equivalent)

.24 x 105 + 4.0 x 105 =

4.24 × 105

2Subtraction works the same

Multiplying in Scientific Notation

This is much easier. Multiply the coefficients and then add the exponents.

(4.0 × 105) × (2.2 × 104) =

8.8 × 109

3

Dividing in Scientific Notation

Divide the two coefficients. Then, subtract their exponents.

(8.8 × 109) / (2.2 × 104) =

4.0 × 105

4

Sample Question

Estimate the potential reduction in petroleum consumption (gallons of gasoline per year) that could be achieved in the United States by introducing electric vehicles under the following assumptions:

• The mileage rate for the average car is 20 miles per gallon of gasoline.

• The average car is driven 15,000 miles per year.

• The United States has 200 million cars.• Twenty percent of the US cars could be

replaced with electric cars.

Sample Question

Start with simplifying and converting the numbers into scientific notation. Then follow the steps outlined in the dimensional analysis section.

 

Sample Question

Now take 20% of your answer

0.20 (2.0 x 10-1) =

Population Growth RateThe most common way to express population growth is as a percentage.

In the equation for population growth rate we subtract deaths from births and divide by the total population then multiply by 100

1

Population Growth RateThe crude birth rate (CBR) is the total number of births per year per 1,000 people and the crude death rate (CDR) is the total number of deaths per year per 1,000 people. When given the CBR and the CDR use the following equation.

2

Sample Question

Electra is tracking its population data. In 1955, the population was 6000, with a crude birth rate of 55. At that time the population of Electra was growing rapidly, because of the low crude death rate of 10. In 1975 the population growth began to slow. The number of deaths totaled 50 and births numbered 120. The total population at the beginning of 1975 was 7000.

Sample Question

What was the population growth rate of Electra in 1955? What was the population growth rate in Electra in 1975?

oo

a)

Why are these 2 calculations different?

The first values were given as crude birth/death rates

Ru

le o

f

To calculate the time required for a doubling of a population based upon population growth rate, PGR, expressed as a percentage. Leave growth percentage as a percentage!

7070707

7070/PGR(leave it

as a percent)= doubling time.

Sample Question

If Electra had maintained the 1955 growth rate how many years would it have taken for the population to double

= 15 ½ years

b)

Per capita is a Latin term that translates into "by head.” It is determined by dividing the total resource by the population.

Per Capita

Sample Question

Calculate the per capita poultry production in 1966 and in 2012.

1966

or 10 kg per capita

2012

or 12 kg per capita

a)

Between 1966 and 2012, the global human population increased from 3.5 billion to 7.0 billion. Global poultry production increased from 35 billion kilograms to 84 billion kilograms during this period.

When analyzing data, contemplate the independent vs. dependent variables. It may help to remember “DRY MIX” to determine which axis to place the variables as you design your graph.

Data, Data Tables and Graphs D

epen

den

t, R

esp

on

din

g o

n t

he Y

-axi

s

Manipulated, Independent on the X- axis

Do not forget that slope is calculated

most simply by change in y (rise) / change in x (run)

Sample Question

Identify the ten-year period during which the greatest increase of the world population growth rate took place.

a)

1960 -1970

Sample Question

How many years did it take for the population in 1960 to double?

b)

40 Years

Sample QuestionThe graphs estimate the Earth’s changing carbon dioxide (CO2) concentration (top) and Antarctic temperature (bottom), based on analysis of ice core data extending back 800,000 years.

Sample Question

Calculate the net change in atmospheric CO2 concentration between 50,000 years ago and today.

a)

80 ppm

Sample Question

Calculate the temperature difference between 550,000 years ago and today.

b)

20 °F

Logarithmic Scale

A scale of measurement that displays the value of a physical quantity using intervals corresponding to orders of magnitude, rather than a standard linear scale. A simple example is a chart whose vertical or horizontal axis has equally spaced increments that are labeled 1, 10, 100, 1000, instead of 1, 2, 3, 4. Examples of logarithmic scales: • Richter (earthquakes)• pH (acids & bases)• decibels (human hearing)

Sample Question

• Determine the threshold(the minimum dose that will produce a detectable effect.)

concentration of pyrethrum.•  Determine the LD50 for

pyrethrum in respect to Daphnia

a) b) c)

1.0 mg/L  400 mg/L