Four biologists are modeling the mass distribution m m\,m of a bacterial culture relative to the radial distance r r\,r from the center of a petri dish. They determine that the rate of change is given by dmdr=1r\displaystyle \frac{dm}{dr} = \frac{1}{r}drdm=r1 for r>0r > 0r>0. Each biologist proposes a different expression for the indefinite integral ∫1r dr\displaystyle \int \frac{1}{r} \, dr∫r1dr:
Dr. Aris: m=lnrm = \ln rm=lnr
Dr. Bond: m=Alnrm = A \ln rm=Alnr
Dr. Chen: m=ln(kr)m = \ln (kr)m=ln(kr)
Dr. Dyani: m=lnr+cm = \ln r + cm=lnr+c
Explain the error in Dr. Aris's proposal.
Explain the error in Dr. Bond's proposal.
Demonstrate why the expressions provided by Dr. Chen and Dr. Dyani are equivalent for certain values of the constants k k\,k and ccc.
