It is possible to unify these two subjects, as some fertilizers are also solid rocket fuel... At 08:08 AM 5/9/2012, Cordwell, William R wrote:
In solid rocket motors, it can be desirable to have a roughly constant burning (surface) area. One solution is to start out with a star shaped hole down the middle; the star erodes as it burns, but the hole also gets larger. Of course, the outside surface is not burning in this case, so it doesn't directly apply to the fertilizer question, but it seems as though you could have a cylinder with an inside hole that gets larger at a similar rate as the outside surface gets smaller.
-----Original Message----- From: math-fun-bounces@mailman.xmission.com [mailto:math-fun-bounces@mailman.xmission.com] On Behalf Of Warren Smith Sent: Wednesday, May 09, 2012 8:58 AM To: math-fun; David Langer; Monty McGovern; Holt, Robert Subject: [EXTERNAL] [math-fun] shapes with constant surface area
The surface of "slow release fertilizer" beads gradually erode, say at 1 centimeter per year. With a spherical bead, the resulting curve of "fertilization rate" would decline with time e.g. it would be proportional to (1-t)^2 for 0<t<1 for a 1cm-radius bead. It might be better to have beads with constant fertilization rate 1.
QUESTION: is there a shape (or set of shapes) whose surface area stays constant as they erode?
The answer in 1 and in 2 dimensions is "yes." In 3D the answer is less obvious, but I have a (presently rather messy) argument the answer also is yes.
More advanced question: you might want the rate to increase with time (as your plants grow bigger they need more)... or be some other curve... WHICH curves F(t) are achievable fertilization-rate curves?
-- Warren D. Smith