Question

The number of bacteria in a culture is increasing according to the law of exponential growth. There are 125 bacteria in the culture after 2 hours and 350 bacteria after 4 hours. (a) Find the initial population. (b) Write an exponential growth model for the bacteria population. Let \(t\) represent time in hours. (c) Use the model to determine the number of bacteria after 8 hours. (d) After how many hours will the bacteria count be \(25,000 ?\)

Short answer

The exponential model is \(P(t) = 86e^{0.51t}\). There will be 1445 bacteria after 8 hours and it will take about 22 hours for the bacteria count to reach 25,000.

Step-by-step solution

Setup the exponential growth model

First, let's assume \(P(t) = P_0 e^{kt}\) is the model for the population at time \(t\), where \(P_0\) is the initial population and \(k\) is the growth rate. We have two points: \(P(2) = 125\) and \(P(4) = 350\). Substituting these points into the exponential growth model gives us two equations, \(125 = P_0 e^{2k}\) and \(350 = P_0 e^{4k}\).

Solve for the growth rate

We divide the second equation by the first to eliminate \(P_0\): \(350/125 = e^{4k} / e^{2k}\), simplifying leads to \(2.8 = e^{2k}\). By taking the natural logarithm of both sides, we get \(\ln(2.8) = 2k\), so \(k = \ln(2.8)/2 \approx 0.51\)

Find the initial population

We substitute the value of \(k\) back into the first equation to find \(P_0\): \(125 = P_0 e^{2*0.51}\). Solving for \(P_0\) gives us \(P_0 \approx 86\).

Write the exponential growth model

With the values of \(P_0\) and \(k\), the exponential growth model becomes \(P(t) = 86e^{0.51t}\)

Determine the number of bacteria after 8 hours

Substitute \(t = 8\) into the model to get the number of bacteria after 8 hours, so \(P(8) = 86e^{0.51 * 8}\), which approximates to \(1445\).

Determine the time for the bacteria count to reach 25,000

To find the time when the bacteria count is 25,000, we set \(P(t) = 25000\) and solve for \(t\). Therefore, \(25000 = 86e^{0.51t}\), which gives \(t = \ln(25000/86)/0.51 \approx 22\) hours.