A biologist is studying a bacterial culture whose mass in milligrams satisfies a…

MATHEMATICS · Multi-part · AP · Calculus AB · Differential Equations · Differential Equations

Problem

A biologist is studying a bacterial culture whose mass B(t)B(t) in milligrams satisfies a differential equation of the form dBdt=kB\dfrac{dB}{dt}=kB, where tt is measured in hours and kk is a non-zero constant. Answer the following questions about the meaning of this model without performing any numerical calculations. (a) Explain what the statement dBdt=kB\dfrac{dB}{dt}=kB conveys about the relationship between the mass of the culture and its instantaneous rate of change. (b) Suppose k>0k>0. Describe how the sign of dBdt\dfrac{dB}{dt} and the size of 1BdBdt\dfrac{1}{B}\dfrac{dB}{dt} compare at t=2t=2 hours and at t=5t=5 hours. (c) If instead k<0k<0, interpret the quantity ∣1BdBdt∣\left|\dfrac{1}{B}\dfrac{dB}{dt}\right| as a percentage and discuss what happens to this percentage over time.

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