Question

Urine culture test is being conducted in the pathology lab from the sample of a patient to prescribe a suitable drug based on bacterial growth. There were 3000 bacteria in the initial sample and it had increased to 3400 in the sample collected after one hour. Based on the exponential growth how long will it take for the bacterial population to reach 10000?

(a) 10 hrs

(b) 8.5 hrs

(c) 9.6 hrs

(d) 3.6 hrs

I got this question in unit test.

The question is from Law of Natural Growth and Decay topic in division Ordinary Differential Equations – First Order & First Degree of Engineering Mathematics

Answer 1

The correct choice is (c) 9.6 hrs

Easiest explanation: As we know according to law of exponential growth, the bacterial population model is

B(t)=ce^kt at t=0 B(0)=ce^0=3000……given –> c=3000 and B(1)=3400=ce^k=3000*e^k

e^k=\(\frac{17}{15}\) taking log,   k=log 17 – log 15 = 0.125, to find t when B(t)=10000 i.e

10000=3000*e^kt = 3000*e^0.125t –> 3.33=e^0.125t –> log 3.33 = 0.125t –> t=9.62 hrs.