Decay Rate

The given problem deals with understanding bacterial growth and decay, followed by the application of an antibacterial treatment. Let's analyze both scenarios using first-order kinetics and decay proportionality.

1. Bacterial Growth (Before Treatment):
   • Bacterial growth follows first-order kinetics, which means the rate of increase in the number of bacteria is directly proportional to the number of bacteria present.
   • Mathematically, this can be expressed as: , where k is the rate constant for growth.
   • The solution of this differential equation for the number of bacteria N over time t is: N = N_0 e^{kt} , where N_0 is the initial number of bacteria.
   • This results in an exponential growth curve.
2. Bacterial Decay (After Treatment):
   • Once the antibacterial medicine is applied, the bacteria begin to decay, and the rate of decay is given as proportional to the square of the existing number of bacteria.
   • This can be represented by: , where r is the decay constant.
   • The solution for this second-order differential equation is: .
   • This implies a hyperbolic decay curve.

The given correct graph: shows an exponential increase followed by a rapid decrease, indicating the bacterial growth and subsequent decay after treatment. Hence, it correctly represents the 'before' and 'after' situations given in the problem.

Other options are incorrect because they either do not match the exponential growth or hyperbolic decay patterns correctly.