How can we use Cardano's method to solve a real life problem?

Question

I am making a math project for my school. We can make it on any topic, but should involve some college level math. I have chosen 'Cardano's method' as my topic. I will be showing the method to solve a general cubic equation$$ax^3+bx^2+cx+d=0$$ using this method and also show how an example with given coefficients a, b, c and d. The problem is I must also be able to solve a real life problem (even if it's hypothetical) using whatever math I am using. At first I considered the real gas equation: $$(P+\frac{an^2}{V^2})(V-nb)=nRT$$ On knowing all the parameters except volume, we get a cubic equation is V. I first tried to plus in some real values for oxygen gas, but the equation I received contained very huge numbers and was difficult to solve. I tried considering hypothetical values of Vander Waal's constant a and b and yet it was pretty difficult to solve using cardano's method. Is there a simple real life problem which requires us to find the roots of a cubic equation and gives simple answers?

Answer 1

I won't give you an explicit question, but I suggest you think of problems like this: A right cylindrical cone has base radius $R$ and height $H$. A right circular cylinder of base radius $r$ and height $h$ is inscribed in the cone. Find its dimensions if its volume is a given fraction $c$ times the volume of the cone.

You can think of other geometric problems with cubes inscribed in spheres, etc.

P.S. I'm not sure I consider this college level math; certainly the basics of complex numbers and DeMoivre's formula shouldn't be.

Answer 2

If you want to solve for $V$ a cubic equation of state, you will face major problems (just as you already noticed).

It is much better to solve it for $Z$, the compressibility factor $Z=\frac{P V}{nRT}$. Then the equation to be solved becomes $$Z^3- \left(1+\frac{b P}{R T}\right)Z^2+\frac{a P}{R^2 T^2}Z-\frac{a b P^2}{R^3 T^3}=0$$

For a real gas, except for very high pressures, the solution will be rather close to $Z=1$ and Newton method will converge as a charm. Do not use Cardano's method.

Now, jus kidding you a little bit, my answer to your question

Is there a simple real life problem which requires us to find the roots of a cubic equation and gives simple answers?

will be : What have I been doing over the last fifty years ?.

If this is of interest for you, search for my papers about cubic equations of state.

Edit

For a first approximation and then as a initial value for Newton method, you can use (for a gas phase) $$Z_0=1+\left(b-\frac{a}{R T} \right)\frac P{R T}$$