Question

The number of bacteria in a culture increased from 27,000 to 105,000 in five hours. When is the number of bacteria one million if:a) Does the number increase linearly with time?b) The number increases exponentially with time?

Answer 1

We have the following situation regarding the growth of bacteria in a culture:

• The given initial population of bacteria is 27,000

,

• After 5 hours, the population increases to 105,000.

Now, we need to find the moment when that population is one million if:

• The population increases linearly with time

,

• The population increases exponentially with time

To find the time in both situations, we can proceed as follows:

Finding the moment when the population is one million if it increases linearly with time

1. We need to find the equation of the line that passes the following two points:

• t = 0, population = 27,000

,

• t = 5, population = 105,000

2. Then the points are:

`\begin{gathered} (0,27000)\rightarrow x_1=0,y_1=27000 \\ (5,105000)\rightarrow x_2=5,y_2=105000 \\ \end{gathered}`

3. Now, we can use the two-point form of the line equation:

`\begin{gathered} y-y_1=(y_2-y_1)/(x_2-x_1)(x-x_1) \\ \\ y-27000=(105000-27000)/(5-0)(x-0) \\ \\ y-27000=(78000)/(5)x=15600x \\ \\ y=15600x+27000\rightarrow\text{ This is the line equation we were finding.} \end{gathered}`

4. We can see that the population is given by y. Then if y = 1,000,000, then we need to solve the equation for x as follows:

`\begin{gathered} 1000000=15600x+27000 \\ \\ 1000000-27000=15600x \\ \\ ((1000000-27000))/(15600)=x \\ \\ x=62.3717948718\text{ hours} \\ \\ x\approx62.3718\text{ hours} \end{gathered}`

Therefore, if the population increases linearly with time, the number of bacteria will be one million around 62.3718 hours.

Finding the moment when the population is one million if it increases exponentially with time

1. In this case, we also need to find the equation that will give us the time when the number of bacteria is one million. However, since the equation will be exponential, we have:

`\begin{gathered} y=a(1+r)^x \\ \\ a\rightarrow\text{ initial value} \\ \\ x\rightarrow\text{ number of time intervals that have passed.} \\ \\ (1+r)=b\text{ }\rightarrow\text{the growth ratio, and }r\rightarrow\text{ the growth rate.} \end{gathered}`

2. Now, we can write it as follows:

`\begin{gathered} a=27000 \\ \\ x=5\rightarrow y=105000 \\ \\ \text{ Then we have:} \\ \\ 105000=27000(b)^5 \\ \end{gathered}`

3. We can find b as follows (the growth factor):

`\begin{gathered} (105000)/(27000)=b^5 \\ \\ \text{ We can use the 5th root to obtain the growth factor. Then we have:} \\ \\ \sqrt[5]{(105000)/(27000)}=\sqrt[5]{b^5} \\ \\ b=1.31209447568 \end{gathered}`

4. Then the exponential equation will be of the form:

`\begin{gathered} y=27000(1.31209447568)^x \\ \\ \text{ To check the equation, we have that when x = 5, then we have:} \\ \\ y=27000(1.31209447568)^5=105000 \end{gathered}`

5. Now, to find the time when the number of bacteria is one million, we can proceed as follows:

`\begin{gathered} 1000000=27000(1.31209447568)^x \\ \\ (1000000)/(27000)=1.31209447568^x \end{gathered}`

6. Finally, we need to apply the logarithm to both sides of the equation as follows:

`\begin{gathered} ln((1000000)/(27000))=ln(1.31209447568)^x=xln(1.31209447568) \\ \\ (ln((1000000)/(27000)))/(ln(1.31209447568))=x \\ \\ x=13.2974595282\text{ hours} \end{gathered}`

Therefore, if the population increases exponentially with time, the number of bacteria will be one million around 13.2975 hours.

Therefore, in summary, we have:

When is the number of bacteria one million if:

a) Does the number increase linearly with time?

It will be 62.3718 hours

b) The number increases exponentially with time?

It will be around 13.2975 hours