# B Problem Solving in Environmental Science

## B.1 Dimensional Analysis

- Length/depth is one dimensional: 1-D (e.g., m)
- Area is two dimensional: 2-D (e.g., m
^{2}) - Volume is three dimensional: 3-D (e.g., m
^{3}) - The dimension of any one unit is related to the exponent of that unit

To convert from one dimension to another, follow the rules for exponents. For example:

- \(length×length=area = L^1 × L^1 =L^{1+1} =L^{2}\)
- \(area×length=volume=L^2 × L^1 =L^{2+1} =L^3\)
- \(volume ÷ area = length= L^3 ÷L^2 =L^{3-2} =L^1\)
- \(volume ÷ length = area = L^3 ÷ L^1 = L^{3-1} = L^2\)

## B.2 Unit Conversions

Measurements are taken in many different units (i.e. metres, miles, inches, feet, pounds, etc.)

ANY unit is converted in one more steps using “conversion factor(s)”

The conversion factor is ALWAYS mathematically equal to ONE (i.e. 1 km ÷ 1000 m = 1) and all components of the conversion factor have an infinite number of significant figures.

Cross multiplication is the key:

~~Given unit~~ x desired unit ÷ ~~given unit~~ = desired unit

### B.2.1 Converting without Dimensional Change

If a woman has a mass of 115 lb, what is her mass in grams? Knowing that 1 lb = 435.6 g, we have:

^{4}g

Two units at the same time: The average speed of a nitrogen molecule in air at 25\(^\circ\)C is 515 m/s. Convert this to miles per hour. Knowing that 1 km = 1000 m, 1 mi = 1.6093 km, 60 s = 1 min, 60 min = 1 hour, we have:

^{3}mi/hr

Converting Across Dimensions:

Earth’s oceans contain approximately 1.36 × 10^{9} km^{3} of water. Calculate the volume in liters. Knowing that 1L=10^{−3}m^{3} and 1km=10^{3} m:

Thus, converting from km^{3} to m^{3} to L, we have: