Thank you for the opportunity to help you with your question!

N'(t) = dN/dt = 4e^(−0.05t) for (0 ≤ t ≤ 20) ∫dN = ∫ 4e^(− 0.05t) dt N = 4[1/(- 0.05)] e^(−0.05t) + C N = - 80 e^(−0.05t) + C when t = 0, N = 65 65 = - 80 e^[(- 0.05)(0)] + C C -80 = 65

C = 145 N(t) = 145 - 80 e^(−0.05t) for (0 ≤ t ≤ 20)

Please let me know if you need any clarification. I'm always happy to answer your questions.