Appendix A. Units of Measurement -- 6 Exercises

1. Find the Errors:
Appendix B shares the following forcefull and usefull advice on units:

when modeling biological or ecological systems .. regularly checking the dimensions in the course of a long calculation provides a rapid check on algebraic accuracy ...a dimensionally correct equation need not be correct, but a dimensionally incorrect equation is invariably nonsense! (Nisbet and Gurney 1982, 21).

Let's put this advice to use--find an error in each of the 9 equations listed below.

variables	their units	Find the Error in each equation:
A	insects	(1) A = C + F*G + T
B	insects/year	(2) A = B*exp(E*T)	note: "exp" stands for the
C	insects	(3) A = C*exp(F)	exponential function
D	dimensionless	(4) A = B*T + F*G + D
E	1/year	(5) D = (A/B) + B*T
F	acres	(6) G = (A+C)/(F+D)
G	insects/acre	(7) B = A*T
T	years	(8) B = (A/F) + (B*T/A)
X	insects/year	(9) X = (A*D)/C

2. Find the error: One of the variables in the flow diagram to the right does not make sense if the variables are measured in the units from the previous exercise. Identify the problematical variable.
3. Verify error free: This diagram is free of obvious errors based on the units listed in the 1st exercise. To verify that the diagram is reasonable, write equations for the flows B and X that are dimensionally correct.

4. Oil Consumption using Metric Units
This exercise is similar to the final exercise in Appendix A. But now we are dealing with gas consumption in Germany rather than in the USA. The stock and flows are taken from Figure 2.8. The oil consumption flow must be measured in BBL/month if time is measured in months and the personal stocks is measured in barrels.

The converters allow us to use metric units that would be familiar to planners in Germany. Also, let's use fuel requirement rather than fuel efficiency since a German driver might refer to his car as requiring, say, 9.5 liters to travel 100 kilometers. The equations are:

oil_consumption = fuel_use*conversion_factor
fuel_use = cars*annual_travel*fuel_requirement

What is the value of the conversion factor if
cars is measured in millions,
annual travel is measured in kilometers per car per year,
fuel requirement is measured in liters of fuel to drive 100 kilometers,
and there are 0.264 gallons in a liter and 42 gallons in a barrel.

5. Measuring River Flows
You may have read about river flows measured in TCFS, thousand cubic feet per second, when you read about The Idagon. Verify that you can multiply river flows measured in million acre-feet per year by 1.38 to obtain flows measured in TCFS. (There are 43,560 cubic feet in an acre-ft, and there are 8,760 hours in a year.)

6. Help Solving the Differential Equation
Chapter 3 describes a differential equation for a population that is likely to grow exponentially over time.
The differential equation is:

dP/dt = rP where

P(t) = population measured in millions of people
P(0) = 10 (the initial population is 10 million people)
r = population growth rate measured in 1/year

Chapter 3 describes a trial and error search for the solution to this differential equation. All but one of the guesses proved to be incorrect because the first derivative of P(t) did not satisfy the differential equation. But you don't need to understand derivatives to rule out many of the guesses. Use your knowledge of units to select the best guess from the list below:

	P(t) = 10 + rt
	P(t) = 10 + rt + rt^2	t^2 is t squared
	P(t) = 10 + exp(t)	exp is the exponential function
	P(t) = 10exp(t)
	P(t) = 10exp(rt)