Units. Session 2 : 9/17

Transcription

1 Units Units allow us to keep track of the meaning of calculations. Every environmental calculation should be accompanied by units. Without them, solutions are meaningless.

2 Units Review: Basic units include time (s, min, h, days, years), mass (g, kg), length (m, cm, nm, mi), temperature ( C, K), etc. Good reference: units.html More complex units (Derived units) can be determined from basic units through equations. F = m a dist. velocity time dist a. = = = 2 time time time dist. kg m F = mass = = N 2 2 time s

3 Examples of Important Units in Environmental Calculations Concentration ( C ) = Mass Flux ( Q ) = mass volume mass area time = = mass length length mass 2 3 time

4 Example Problems A truck spills 2m 3 of 5kg/m 3 parathion, a highly toxic organophosphate insecticide, into a soil region that has dimensions of 5m x 6m x 90cm. After 5 days, two regions of the contaminated area are tested, and the concentration of each is found to be 1.3kg/m 3 and 4.0kg/m 3. A. What is the total mass of parathion spilled? B. If the parathion was distributed evenly throughout the contaminated area, what would be the concentration? C. Write an inequality to describe the range of parathion concentration within the contaminated area. D. Assuming that the actual concentration within the initially contaminated zone is the average of the two test sites, calculate the mass of parathion that has been transported out of the initial contamination zone.

5 The Cartesian Plane Rectangular Coordinate System y axis y positive x negative x positive x axis y negative Origin (0,0)

6 Why is it important to be able to graph? Graphing allows us to quickly and simply assess trends which would be less obvious if represented numerically.

7 Describing Points on a Graph In labeling points: x term first, y term second y axis (2,3) **Axes can have any units, but must be clearly defined. ( 3, 2) ( 1, 2) (3, 2) x axis 3

8 Scatter Plots Plotting multiple points to describe an equation or a trend. Example: Plot the following data given for a wolf population Year WP Adult Wolf Population Year Plescher et al., J. Wildl. Manage. 61 (2), 1997

9 Example: Bacterial Growth ESEM Images: Biofilm exposed to Uranium Biofilm Growth Curve OD (600nm) Elapsed Time (h)

10 Distance between two points Distance between points (x 1,y 1 ) and (x 2,y 2 ) : y axis D = ( x y x 1 ) + ( y 2 1 ) D (x 1,y 1 ) (x 2,y 2 ) x axis

11 Environmental Example: Contaminant Transport A restoration team is tracking a plume of the herbicide atrazine that has seeped into groundwater near a processing plant. To track the chemical, the team has made a grid of the soil as shown below. In May (shown in pink), the leading edge of the plume was located 1m North and 1m West of the center of the team s grid. In November, the team finds that the leading edge of the plume had reached 2m North and 3m east of the center of the team s grid. Question: How far had the leading edge of the atrazine plume travelled between May and November? Note: Gridlines on each axis represent 1m intervals.

12 Midpoint Formula Midpoint between points (x 1,y 1 ) and (x 2,y 2 ): x1 + x 2 y 1 + y 2 Midpoint =, 2 2

13 Graphs of Equations Sketching by point plotting: Plug in x values to find y (or vise versa), plot

14 Finding Intercepts Intercepts are points at which the graph intersects the x or y axis. y axis x intercept occurs when y=0 x axis y intercept occurs when x=0

15 Other scenarios: Not all graphs have a y intercept, x intercept or either no x or y intercept y axis One x and one y intercept One x intercept, No y intercept x axis

16 Solving for Intercepts of an Equation: To find x intercepts, let y be zero and solve for x To find y intercepts, let x be zero and solve for y

17 Equation of a Circle Standard Form: ( x h ) + ( y k ) = r Represents a circle centered at point (h,k) with radius r

18 Points of Intersection A point of intersection between two graphs is a point that satisfies both functions To find the points of intersection, set two y values equal to eachother and solve the equation

19 Example: A study of fish populations is conducted in two different streams: one that is contaminated with DDT and has a declining fish population, and one pristine stream that was stocked three years prior. Ecologists have determined the following equations to describe the two populations: 1. Contaminated Stream: P = 260 4t 2 2. Pristine Stream: P = 3t Where P = fish Population, and t = elapsed time in years. Question: How long will it take for the population in the pristine stream to equal that of the contaminated stream?

20 Graphed Solution: 300 Fish Population Contaminated Stream Pristine Stream Years

21 Slope-Intercept Form Simplest mathematical model to describe a linear relationship between two variables y = mx + b SLOPE y intercept y variable x variable

22 Slope: What is it and how do we find it? Slope: Vertical Change per Unit of Horizontal Change y axis x (x 2,y 2 ) m y y y y = = 2 1 x x x x 2 1 (x 1,y 1 ) axis Horizontal Line: Slope = 0 Vertical Line: Slope = undefined!

23 Slope in an Environmental Problem Slope can also be used to represent a rate, depending on the units of your coordinate system Calculate average rate of CO 2 increase per year assuming that in 2004, CO 2 concentration was 377ppm, and in 2008, 384ppm

24 Functions A function is a relationship between two variables such that to each value of the independent variable, there is exactly one value of the dependent variable. Function notation: Generally, isolate the dependent variable on the left In function notation, the dependent variable (e.g. y) can be written as f(x). Evaluating Functions

25 Combinations of Functions Multiple functions can be combined to create new functions e.g. f(x) + g(x), f(g(x))

26 Limits If a function f(x) becomes arbitrarily close to a single value L as x approaches c from either side, then lim f ( x ) = L x c In many instances (for continuous functions), the limit L can be determined by substituting c into the function