Three Essential Mathematical Skills

More often than not when a search-and-rescue team finds a hiker dead in the wilderness, it turns out that the person was not carrying some item from a short list of essentials, such as water and a map. There are three mathematical essentials in this course.

1. Converting units

basic technique: section 0.9, p. 30; conversion of area, volume, etc.: section 1.1, p. 41
Examples:

To check that we have the conversion factor the right way up (10³ rather then 1/10³), we note that the smaller unit of grams has been compensated for by making the number larger.For units like m^2, kg/m³, etc., we have to raise the conversion factor to the appropriate power:

Examples with solutions — p. 37, #6; p. 59, #10 Problems you can check at lightandmatter.com/area1checker.html — p. 37, #5; p. 37, #4;p. 37, #7; p. 59, #1; p. 60, #19

2. Reasoning about ratios and proportionalities

The technique is introduced in section 1.2, p. 43, in the context of area and volume, but it applies more generally to any relationship in which one variable depends on another raised to some power.Example: When a car or truck travels over a road, there is wear and tear on the road surface,which incurs a cost. Studies show that the cost per kilometer of travel C is given by

where w is the weight per axle and k is a constant. The weight per axle is about 13 times higher for a semi-trailer than for my Honda Fit. How many times greater is the cost imposed on the federal government when the semi travels a given distance on an interstate freeway?.

First we convert the equation into a proportionality by throwing out k, which is the same for both vehicles:

Next we convert this proportionality to a statement about ratios:

Since the gas taxes paid by the trucker are nowhere near 29,000 times more than those I pay to drive my Fit the same distance, the federal government is effectively awarding a massive subsidy to the trucking company. Plus my Fit is cuter.532 Chapter 20 Bounded waves

Examples with solutions — p. 59, #11; p. 59, #12; p. 60, #17; p. 119, #16; p. 120, #22; p. 252,#6; p. 276, #10; p. 277, #15; p. 279, #19; p. 305, #8; p. 305, #9 Problems you can check at lightandmatter.com/area1checker.html— p. 60, #16; p. 60,#18; p. 61, #23; p. 62, #24; p. 62, #25; p. 199, #7; p. 252, #8; p. 275, #5; p. 276, #8; p. 279,#21; p. 304, #4; p. 415, #2

3. Vector addition

section 7.3, p. 208

Example: The ?r vector from San Diego to Los Angeles has magnitude 190 km and direction 129?counterclockwise from east. The one from LA to Las Vegas is 370 km at 38?counterclockwise from east. Find the distance and direction from San Diego to Las Vegas.

Graphical addition is discussed on p. 208. Here we concentrate on analytic addition, which involves adding the x components to find the total x component, and similarly for y. The trig needed in order to find the components of the second leg (LA to Vegas) is laid out in figure eon p. 205 and explained in detail in example 3 on p. 205:

(Since these are intermediate results, we keep an extra sig fig to avoid accumulating too much rounding error.) Once we understand the trig for one example, we don’t need to reinvent the wheel every time. The pattern is completely universal, provided that we first make sure to get the angle expressed according to the usual trig convention, counterclockwise from the x axis.Applying the pattern to the first leg, we have:

For the vector directly from San Diego to Las Vegas, we have

The distance from San Diego to Las Vegas is found using the Pythagorean theorem,

(rounded to two sig figs because it’s one of our final results). The direction is one of the two possible values of the inverse tangent

Consulting a sketch shows that the first of these values is the correct one.

Examples with solutions — p. 232, #8; p. 232, #9; p. 381, #8

Problems you can check at lightandmatter.com/area1checker.html — p. 214, #3; p. 214,#4; p. 231, #1; p. 231, #3; p. 234, #16; p. 275, #3; p. 280, #23; p. 380, #3