Answer: a) [tex]N(t) = 10^3\exp(0.046\frac{1}{min}t)[/tex]
b) 1,000,000 bacteria at t = 150 min
Step-by-step explanation:
Hi!!
A colony that grows exponentially has a number of bacteria:
[tex]N(t) = N_0 \exp(\lambda t)[/tex]
In this case at time t = 0:
[tex]N(0)=N_0=10^3[/tex]
We need to find the value of λ. We use the data:
[tex]N(t=50\;min)10^4 = 10^3\exp(\lambda \;50\;min)[/tex]
[tex]ln(10)=2.3=\lambda\;50\;min\\\lambda= \frac{0.046}{min}\\N(t) = 10^3\exp(\frac{0.046}{min}t)\\[/tex]
To find when there will be 1,000,000 bacteria:
[tex]10^6=10^3\exp(\frac{0.046}{min}t)[/tex]
[tex]\ln(10^3)=3\ln(10) = \frac{0.046}{min}t[/tex]
[tex]t = 150\;min [/tex]
