Cardano and Bombelli: No big difficulty here. I asked for two good solutions to be posted, one working from the formula given in the text and one from the algorithm that led to the formula. (See also the following remark.)
Trisection: Here the example given in the text for Bombelli's method was misleading. In general the real part of the cube will not be the same as the real part of the cube root. Bombelli (back in the previous problem) apparently guessed that the real part of the two cube roots would be 3/2 by observing that their sum must be 3 -- remember that he knew this answer to the cubic equation beforehand and merely wanted to rationalize it from the Cardano formula -- and had some kind of confused intuition that the two terms were complex conjugates.
Back to the trisection equation. To analyze it I had to return to the formula for the solutions of the cubic given in the National Bureau of Standards Handbook (ed. by Abramowitz and Stegun), which I'll summarize here for the special case where the quadratic term is zero. (Forced to lapse into TeX here.)
Logarithmic data distribution: As many of you noticed, this doesn't always work precisely as advertised. I think there are two main reasons for that:
Rule of 72: Everybody chose to do this one; past a certain age, pensions have more resonance than wildlife ecology. Most everybody saw that the relevant formula is
t = ln(2)/ln(1+r),
where 100r = R is the interest rate in percent. Since ln(2) = .69... and ln(1+r) is approximately r for small r, we have t approximately 69/R. So, why 72 instead of 69? For one thing, 72 has lots of divisors, so it makes the back-of-the-envelope calculations come out even. More quantitatively, note the Taylor expansions
ln(1+r) = r - r2/2 + ... and hence
1/ln(1+r) = 1/r [1 + r/2 + ...].
For a typical interest rate like 6%, 1 + r/2 = 1.03. And 69 x 1.03 is not 72, but it's getting there! No number in the numerator will work exactly for all interest rates, but a number slightly greater than 69 will do a decent job for any plausible interest rate.
The first 4 were straightforward. I think we and our students are all happy that "subtangent" and "subnormal" have disappeared from the syllabus.
De Sluse: This topic should be X-rated: no high school algebra students allowed to see the calculations! When you look closely, you see that de Sluse is just doing what our textbooks now call implicit differentiation, but in a horribly ambiguous and confused notation.