Question

The population of the bacteria colony starts at 100 and grows by $30\%$ per hour.
A.Find the formula for the number of bacteria, P, after t hours.
B.What is the doubling time of this population; that is, how long does it take the population to double in size?

Answer 1

Hint : To solve this question, we need to use the concept of exponential growth. The exponential growth occurs when the growth rate of the value of a mathematical function is proportional to the function’s current value, resulting in its growth with time being an exponential function. We can also say that when the growth of a function increases rapidly in relation to the increase in the total number, then it is exponential.
Formula used:
Systems that experiences exponential growth increase according to the mathematical model:
$P = {P_0}{e^{kt}}$
Where, ${P_0}$ represents the initial state of the system and $k$ is a constant, called the growth constant.

Complete step by step solution:
We are given that the bacteria colony starts at 100 and grows by $30\%$ per hour.
Therefore, ${P_0} = 100$ and $k = 0.3$ .
By using the formula $P = {P_0}{e^{kt}}$ , we can say that $P = 100{e^{0.3t}}$ .
Thus, the answer for our first question is: the formula for the number of bacteria, P, after t hours is: $P = 100{e^{0.3t}}$ .
We also know that the continuous growth rate is nothing but the growth constant.
Therefore, the answer to the third question is: the continuous growth rate for the colony is $0.3$ .
Now, for the second question, we have to find the doubling time.
Therefore, $P = 2{P_0} = 200$ . Putting this value into the main equation, we will get

$$200 = 100{e^{0.3t}} \\\ \Rightarrow 2 = {e^{0.3t}} \;$$

Now, we will take logarithms on both the sides.

$$\Rightarrow \ln 2 = \ln {e^{0.3t}} \\\ \Rightarrow \ln 2 = 0.3t\ln e \\\ \Rightarrow 0.693 = 0.3t \\\ \Rightarrow t = 2.31 \,hours \;$$

Thus, our second answer is: The doubling time of the population is $2.31 \, hours$ .

Note : Here, in this question, we have used two rules of the logarithms. First, the exponent rule: $\ln {a^b} = b\ln a$ . And the second rule which states that the logarithm of the number with the same base is 1 that is $\ln e = {\log _e}e = 1$ .