Question

A crop initially has N₀ Bacterias. After 1 hour the crop has reached ( 3/2 ) N₀ Bacterias. If the rapid growth in that crop is proportional to the number bacteria present at the time t, determine the time required for the number of bacterias grouper are tripled.

Answer 1

Answer:

2.71 hours are required for the number of bacterias grouper are tripled.

Step-by-step explanation:

The number of bacterias can be given by the following exponential function:

$N(t) = N_{0}e^{rt}$

In which $N(t)$ is the number of bacterias at the time instant t, $N_{0}$ is the initial number of bacterias and r is the rate for which they grow.

After 1 hour the crop has reached ( 3/2 ) N₀ Bacterias.

This means that $N(1) = 1.5N_{0}$. With this information, we can find r.

$N(t) = N_{0}e^{rt}$

$1.5N_{0} = N_{0}e^{r}$

$e^{r} = 1.5$

To find r, we apply ln to both sides

$\ln{e^{r}} = \ln{1.5}$

$r = 0.405$

Determine the time required for the number of bacterias grouper are tripled.

This is t when $N(t) = 3N_{0}$

$N(t) = N_{0}e^{0.405t}$

$3N_{0} = N_{0}e^{0.405t}$

$e^{0.405t} = 3$

Again, we apply ln to both sides

$\ln{e^{0.405t}} = \ln{3}$

$0.405t = 1.10$

$t = 2.71$

2.71 hours are required for the number of bacterias grouper are tripled.