tweaks

quarto/precalc/calculator.qmd

@@ -190,32 +190,46 @@ A right triangle has sides $a=11$ and $b=12$. Find the length of the   hypotenus
 
 ##### Example
 
-An article in the [Washington Post](https://www.washingtonpost.com/climate-environment/2022/09/19/ants-population-20-quadrillion/) describes estimates for the number of ants on earth.
+A overview of a research paper published  in [theconversation.com](https://theconversation.com/earth-harbours-20-000-000-000-000-000-ants-and-they-weigh-more-than-wild-birds-and-mammals-combined-190831) reviews six authors' work on estimating the number of ants currently on earth. This was covered in an
+article in the [Washington Post](https://www.washingtonpost.com/climate-environment/2022/09/19/ants-population-20-quadrillion/).
 
 
-They describe the number of ants two ways:
+The authors describe the number of ants two ways:
 
-* There are $20$ *trillion* ants.
-* There are $12$ megatons of ants
+* There are $20 \cdot 10^{15}$ (20 *quadrillion* or 20 thousand million millions) ants.
 
-With this how many ants make up a pound?
+* There is an estimated total biomass of $12$ megatons of *dry* carbon.
 
-Below we use the underscore as a separator, which is parsed as commas are to separate groups of numbers. The calculation is the number of ants divided by the number of pounds of ants (one megaton is $1$ million pounds):
+
+The authors note in a supplement to their paper that over 15,700 species and subspecies of ants have been named. To get a good estimate, 489 studies of ant populations were combined spanning all continents and major habitats. The studies identify the number of *foraging* ants and combined yield the estimates for the *epigaeic ant abundance* ($3.02 \cdot 10^{15}$) and the *arboreal ant abundance* ($1.34\cdot 10^{15}$). Using an estimate of 22% of ants in a colony are foraging, they get the following estimate for the number of ants:
 
 ```{julia}
-20_000_000_000_000_000 / (1_000_000 * 12 * 2000)
+(3.02 * 10^15 + 1.34*10^15) * 100 / 22
 ```
 
-Or not quite a million per pound.
-
-Using a pound is $2.2$ killograms or $2,200$ grams, we can this many ants per gram:
+Shifting the decimal point, this gives a value rounded to $20\cdot 10^{15}$ ants.
 
+The authors used a value for the *dry weight* of an average (and representative) single ant. What was that value? (Which they indicate is perhaps unreliable,
+as, for example, small-bodied ants may be much more abundant than large-bodied ants).  We assume below that one "megaton" is $1$ million *metric* tons; a metric ton is $1,000$ kilograms; and a kilogram $1,000$ grams:
 
 ```{julia}
-20_000_000_000_000_000 / (1_000_000 * 12 * 2000) / 2200
+(12 * 1_000_000 * 1_000 * 1_000) / 20_000_000_000_000_000
 ```
 
-Such combinations will be easier to check for correctness when variable names are assigned the respective values.
+Which translates to an *average* dry *carbon* weight of $0.6/1000$ grams, that is $0.6$ milligrams ($0.62$ mg C was actually used).
+
+ The authors write that insects are generally considered to have a dry weight of 30% wet weight, and a carbon weight of 50% dry weight, so the weight in grams of an *average* living ant would be multiplied by $2$ and then $10/3$:
+
+```{julia}
+(12 * 1_000_000 * 1_000 * 1_000) / 20_000_000_000_000_000 * (2 * 10/3)
+```
+
+That is 4 milligrams, or 250 ants per gram on average.
+
+Numeric combinations, as above, will be easier to check for correctness when variable names are assigned to the respective values.
+
+Using the underscore, as above, to separate groups of digits, is helpful, as an alternate to scientific notation, when working with large numbers.
+
 
 
 
@@ -1094,7 +1108,21 @@ answ=1
 radioq(choices, answ)
 ```
 
+##### Question
 
+[Sagan's](https://en.wiktionary.org/wiki/Sagan%27s_number) number is defined to be the total number of stars in the observable universe. How big is it? A *sextillion* is 7 groups of three 0's after a leading 1. One estimate is $10$ sextillion. How might this be entered into `Julia`? Select the one that *doesn't* work:
+
+```{julia}
+choices = [
+"`10*10^21`",
+"`10*10.0^21`",
+"`10e21`",
+"`1e22`",
+    "`10_000_000_000_000_000_000_000`"]
+explanation = "With an *integer* base, `10^21` overflows. For typical integers, onmly `10^18` is defined as expected. Once `10^19` is entered the mathematical value is larger the the `typemax` for `Int64` and so the value *wraps* around. The number written out with underscores to separate groups of 0s is parsed as an integer with 128 bits, not 64."
+buttonq(choices, 1; explanation=explanation)
+
+The estimate of 10 sextillion for Sagan's number was made in 1980, a more modern estimate is 30 times larger.
 
 ##### Question
 

quarto/precalc/polynomial.qmd

@@ -459,6 +459,13 @@ plot!(x -> -x^4, -3,3, legend=false, xticks=false, yticks=false, subplot=3, titl
 plot!(x -> -x^5, -3,3, legend=false, xticks=false, yticks=false, subplot=4, title="n > odd, aₙ < 0")
 ```
 
+##### Example
+
+This graphic shows some of the above:
+
+[![you tube](https://img.youtube.com/vi/OFzqDatEvCo/3.jpg)](https://m.youtube.com/watch?v=OFzqDatEvCo)
+
+
 ##### Example
 
 

quarto/precalc/variables.qmd

@@ -111,6 +111,54 @@ a = v0 * cosd(theta)
 By defining a new variable `a` to represent a value that is repeated a few times in the expression, the last command is greatly simplified. Doing so makes it much easier to check for accuracy against the expression to compute.
 
 
+##### Example
+
+A [grass swale](https://stormwater.pca.state.mn.us/index.php?title=Design_criteria_for_dry_swale_(grass_swale)) is a design to manage surface water flow resulting from a storm. Swales detain, filter, and infiltrate runoff limiting erosion in the process.
+
+![Swale cross section](precalc/swale.png)
+
+There are a few mathematical formula that describe the characteristics of swale:
+
+The area is given by:
+
+$$
+A = (b + d/\tan(\theta)) d
+$$
+
+The *wetted* perimeter is given by
+
+$$
+P = b + 2 d/\sin(\theta)
+$$
+
+The *hydraulic radius* is given by
+
+$$
+R = \frac{b\cdot d \sin(\theta) + d^2 \cos(\theta)}{b\sin(\theta) + 2d}.
+$$
+
+Finally, the *flow quantity* is given by *Manning's* formula:
+
+$$
+Q = vA = \frac{R^{2/3} S^{1/2}}{n}, \quad R = \frac{A}{P}.
+$$
+
+With $n$ being Manning's coefficient, $v$ the velocity in feet per second, and $S$ being the slope. Velocity and slope are correlated.
+
+Manning's coefficient depends on the height of the vegetation in the grass swale. It is $0.025$ when the depth of flow is similar to the vegetation height.
+
+Given all this, compute the flow quantity when $S = 2/90$, $n=0.025$ and $v=1/10$ for a swale with characteristics $b=1$, $\theta=\pi/4$, $d=1$.
+
+```{julia}
+b, theta, d = 1, pi/4, 1
+n, S = 0.025, 2/90
+A = (b + d/tan(theta)) *  d
+P = b + 2d/sin(theta)
+R = A / P
+Q = R^(2/3) * S^(1/2) / n
+```
+
+
 ##### Example