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December 7th, 2017, 06:36 PM  #1 
Newbie Joined: Jan 2014 Posts: 17 Thanks: 0  unit of measurement
We are given the following model for population growth $\displaystyle \frac{dP}{dt}=kP$ where P(t) is the number of population at time t (measured in years) and k is a constant of proportionality. In this model the unit for time is year. What is the unit for the constant of proportionality? per year? how to write it in symbol? year$\displaystyle ^{1}$? 
December 7th, 2017, 06:40 PM  #2 
Senior Member Joined: Sep 2016 From: USA Posts: 383 Thanks: 206 Math Focus: Dynamical systems, analytic function theory, numerics 
It is dimensionless. The equation is "unit free" in the sense that the object on the left is a rate of change and the object on the right is a population. This equation doesn't say that these objects are equal or that they have the same units. It says the the rate of change (just the scalar value, not the unit) is equal to k*P (also just the scalar, not the unit).

December 8th, 2017, 02:36 AM  #3 
Math Team Joined: Jan 2015 From: Alabama Posts: 3,191 Thanks: 871 
I disagree. If P is a "population", so is measured in "people" (or "millions of people") and t is in years, then dP/dt is the rate of change of population in "people per year" (or "millions of people per year") so that k must be in "per year" as you say. Yes, that can be written as "$\displaystyle year^{1}$".

December 14th, 2017, 08:15 AM  #4 
Senior Member Joined: Apr 2014 From: Glasgow Posts: 2,111 Thanks: 706 Math Focus: Physics, mathematical modelling, numerical and computational solutions 
Yeah... "per year", "/year" or "year$\displaystyle ^{1}$" is fine.
Last edited by Benit13; December 14th, 2017 at 08:18 AM. 
December 20th, 2017, 01:04 AM  #5 
Newbie Joined: Jan 2014 Posts: 17 Thanks: 0 
Let $\displaystyle T(t)$ (measured in $\displaystyle ^\circ{C}$) be the temperature of a cup of coffee at time $\displaystyle t$ (measured in minutes). Newton's law of cooling gives the differential equation $\displaystyle \frac{dT}{dt}=k(TT_m)$, where $\displaystyle T_m$ (measured in $\displaystyle ^\circ{C}$) is the ambient temperature and $\displaystyle k$ is a constant of proportionality. If $\displaystyle T_m=25^\circ C$, the solution of the differential equation is $\displaystyle T(t)=25+ce^{kt}$, where $\displaystyle c$ is an arbitrary constant. Does $\displaystyle c$ have a unit? If $\displaystyle c$ has a unit, then what is the unit of $\displaystyle c$?

December 21st, 2017, 05:45 AM  #6 
Math Team Joined: Jan 2015 From: Alabama Posts: 3,191 Thanks: 871 
The exponential of a unit is not defined so, first, since t has units of "minutes", k must have units of "minute$^{1}$" or "1/minutes". That way, "kt" is "unitless" and $e^{kt}$ is "unitless". Since both "T" and "25" have units of "degrees Celcius", c must have units of "degrees Celcius". 

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