Question

Given two points for an exponential function,

1. Use the two points to find the growth rate, k. Write an exponential model for each point, then solve this system of two equations for k.
2. Use either point with the k you found to find the initial amount at time zero, A..
3. Doubling time is when the amount is 2 A
4. Use the values of k and A to calculate the amount for a given time or to find the time to reach a specific amount in the future.
The count in a bacteria culture was 100 after 15 minutes and 1200 after 30 minutes. Assuming the count grows exponentially, What was the initial size of the culture?

Answer 1

To find the initial size of the bacterial culture, we use the exponential growth model N(t) = A * e^(kt) with two given data points to solve for the growth rate k and the initial amount A. The initial size of the culture is approximately 8.33 bacteria.

Step-by-step explanation:

To solve for the initial size of the bacterial culture, we start by writing an exponential growth model using the formula N(t) = A * e^(kt), where N(t) is the count at time t, A is the initial amount, and k is the growth rate.

We have two points: (15, 100) and (30, 1200), which give us the system of equations:

1. 100 = A * e^(15k)
2. 1200 = A * e^(30k)

Divide the second equation by the first to find k:

1200 / 100 = e^(30k) / e^(15k)

12 = e^(15k)

ln(12) = 15k

k = ln(12) / 15

Now that we have k, plug it back into the first equation to find A:

100 = A * e^((ln(12) / 15) * 15)

100 = A * e^(ln(12)) = A * 12

A = 100 / 12

A = 8.33

Therefore, the initial size of the culture was approximately 8.33 bacteria.