DE equation modeling growth off a tan function

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SUMMARY

The discussion focuses on formulating a differential equation (DE) to model bacterial growth based on the tangent function. The user correctly identifies that the growth rate is proportional to the tangent function, leading to the equation dP/dt = k tan(t) for bacteria growth up to 1.5 mg. Beyond this threshold, the growth rate is defined piecewise, with dP/dt constrained between π/6 mg and π/3 mg per hour. This approach effectively captures the behavior of the bacterial population as it approaches and surpasses the specified limit.

PREREQUISITES
  • Understanding of differential equations, specifically growth models.
  • Familiarity with the tangent function and its properties.
  • Knowledge of piecewise functions and their applications in modeling.
  • Basic concepts of bacterial growth dynamics and proportionality.
NEXT STEPS
  • Study the derivation of differential equations for biological growth models.
  • Learn about piecewise functions and their implementation in mathematical modeling.
  • Explore the properties of the tangent function and its applications in growth scenarios.
  • Investigate the implications of growth rate thresholds in population dynamics.
USEFUL FOR

Students in biology or mathematics, particularly those studying population dynamics, differential equations, and mathematical modeling of growth processes.

Outlaw747
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Homework Statement


Data suggests rate of growth of bacteria is proportional to the tangent function evaluated on the amount of bacteria present at time t, up to 1.5 mg. After 1.5 mg, value maintained. pi/6 to pi/3 mg growth in 1 hour. How would I write a DE equation based on this? I'm not asking how to solve it but to set the equation up.


Homework Equations


dP/dt = kP
P(t) = P-initial * e^(kt)


The Attempt at a Solution


I am not really sure how to go about this. If we use the second equation isn't that based off another growth model? If I have dP/dt = tan (t) , that doesn't really make sense. Just need a push in the right direction if possible. Tried a few different attempts but none really worked.
 
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The question stated "is proportional to", not "equals", so that should be dP/dt = k tan(t). That's only until 1.5mg though. If I understand correctly, after 1.5 mg, P increases by at least pi/6 mg, and at most pi/3 mg, each hour? For that part I would suggest something like a piecewise definition- Before 1.5mg, dP/dt = k tan(t), after 1.5mg, [tex]\pi/6 < dP/dt < \pi/3[/tex].
 
K I'll try that, thanks.
 

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