Question

The population of a culture of bacteria, P(t), where t is time in days, is growing at a rate that is proportional to the population itself and the growth rate is 0.2. The initial population is 10.(1) What is the population after 50 days? (Do not round your answer.)

Answer 1

The population of the bacteria culture after 50 days can be calculated using the exponential growth formula P(t) = P0 * e^(rt), where P0 is the initial population, r is the growth rate, and t is the time in days. For an initial population of 10 and a growth rate of 0.2, the population after 50 days is found by evaluating P(50) = 10 * e^(0.2 * 50).

Step-by-step explanation:

The student is asking how to calculate the population of a culture of bacteria after a given time period, given that the population grows exponentially at a constant rate. This is a problem involving exponential growth, which can be solved using the formula for exponential functions:

P(t) = P0 ert

Where:

• P(t) is the population at time t,
• P0 is the initial population,
• r is the growth rate,
• e is the base of the natural logarithm (approximately 2.71828),
• t is the time in days.

In this case, the initial population P0 is 10, the growth rate r is 0.2, and we want to find P(50), the population after 50 days. Plugging these values into the equation we get:

P(50) = 10 * e(0.2 * 50)

Calculating this expression will give us the population after 50 days.

Answer 2

Okay, here we have this:

Considering the provided information, we are going to calculate the requested population, so we obtain the following:

Then we will substitute in the following exponential growth formula:

`n(t)=n_0(1+r)^t`

Then, replacing:

`\begin{gathered} n(50)=10(1+0.2)^(50) \\ n(50)=10(1.2)^(50) \\ n(50)\approx91004.38 \end{gathered}`

Finally we obtain that the population will be approximately 91004.38 after 50 days.