Algebraic Modelling
Find the shape, area, and volume of an algebraic lake?
In the land of Trianon the land surface is modeled exactly by the algebraic surface
z = x^3 + yx^2 + x^2 – xy^2 – x + y^2 – y, where the positive z-axis points vertically upward.
When it rains a lake is formed in Trianon, which eventually becomes as full as possible.
When the lake is maximally full,
a) What is the shape of the lake?
b) What is the surface area of the lake? (i.e. the surface area of the planar top of the lake)
c) What is the volume of the lake?
a) Anyone who’s ever fooled around with cubic surfaces might guess that the shape of the lake will be triangular, the lines forming it found by factorizing x^3 + yx^2 + x^2 – xy^2 – x + y^2 – y – 1 = (1-x) (y-(1/2)(1-√5)(1+x)) (y-(1/2)(1+√5)(1+x)) = 0.
b) The surface area of this triangle is 2√5 = 4.47214, made convenient by the fact one of the lines is x = 1. The 3 vertices are at (-1,0), (1,1+√5), (1,1-√5).
c) This was a much more of a hassle, but the volume comes out to (10/3)(25 – 11√5) = 1.34417.
Edit: An example of a cubic surface that has such a lake that’s not triangular in shape would be (y – x²)(y -1). There aren’t too many different categories for cubic surface “lakes”.
Edit 2: To be pendantic, one can find the saddle points by differentiating the equation with respect to both x and y, and solving the simultaneous set of equations, both equal to 0. In this way, the 3 saddle points for this particular equation can be found in addition to the minimum inside the “lake”. The lowest of the 3 saddle points would be the maximum lake surface elevation. The depth of the lake is 40/27 at (1/3,2/3).
Finding Instantaneous Rate of Change: first Proncipals Algebraic Model
