Math Question Exponential and Logarithmic Models

Question

A bacteria culture initially contains 2500 bacteria and doubles every half hour. 

Find the size of the bacterial population after 40 minutes.

Find the size of the bacterial population after 6 hours.

Answer 1

Math Question Exponential and Logarithmic Models: A bacteria culture initially contains 2500 bacteria and doubles every half hour. Find the size of the bacterial population after 40 minutes. Find the size of the bacterial population after 6 hours.

Detailed Solution:

Given/Known: P(t) = P(0) e^(kt) P(0) = 2500 P(t) = 2500 e^(kt) P(30) = 2P(0)

a) P(40) = ?? b) P(6*60) = P(360) = ??

P(30) = P(0) e^(k*30)

2P(0) = P(0) e^(k*30)

2 = e^(k*30)

Ln (2) = Ln (e^(k*30)

Ln 2 = (k*30) Ln (e) = 30k

k = (Ln 2 / 30)

P(t) = P(0) e^(kt)

P(t) = 2500 e^(kt)

P(t) = 2500 e^{(Ln 2 / 30) * (t)}

a) P(40) = 2500 e^{(Ln 2 / 30) * (40)} = 2500 e^{(40)*(Ln 2 / 30)} = 2500 e^{(4/3)Ln 2} (Exact) = 6300 (Appeox)

b) P(6*60) = P(360) = 2500 e^{(Ln 2 / 30) * (360)} = 2500 e^{(360)*(Ln 2 / 30)} = 2500 e^{12 Ln 2} (Exact) = 1,0240,000 (Appeox)

Answer 2

A(t) = a₀(2)^t/30

A(40) = 2500(2)^40/30

A(360) = 2500(2)^360/30, 6 hours = 360 minutes