Question

While growing bacteria in a laboratory Petri dish, a scientist notices the number of bacteria over time. The observations are shown in the table.
Number of bacteria
Number of minutes
from initial state
0 4
5 128
10 4096
15 131072

What function can be used to describe the number (n) of bacteria after 2 minutes?

Answer 1

Answer:

The function that can be used to describe the number (n) of bacteria after 2 minutes is;

$P = 4 \cdot e^{\left(\dfrac{ln(32)}{5} \times 2\right)} \approx 4 \cdot e^{\left(0.693\times 2\right)}$

Step-by-step explanation:

The data in the table are presented as follows;

Number of bacteria; 4, 128, 4,096, 131,072

Number of minutes from initial state; 0, 5, 10, 15

The general equation for population growth is presented as follows;

$P = P_0 \cdot e^{r\cdot t}$

Where;

P = The population after 't' minutes

P₀ = The initial population

r = The population growth rate

t = The time taken for the growth in population numbers

At t =  minutes. we have;

$4 = P_0 \cdot e^{r\times0} = P_0$

∴ P₀ = 4

At t = 5, we have;

$128 = 4 \cdot e^{r\times 5}$

$\therefore e^{r\times 5} = \dfrac{128}{4} = 32$

$ln\left(e^{r\times 5}\right) = ln(32)$

∴ r × 5 = ㏑(32)

r = ln(32)/5 ≈ 0.693

The number (n) of bacteria after 2 minutes is therefore;

$P = 4 \cdot e^{\left(\dfrac{ln(32)}{5} \times 2\right)} \approx 4 \cdot e^{\left(0.693\times 2\right)}$