A bacteria culture starts with 1,000 bacteria and doubles in size every 2 hours. Find an exponential model for the size of the culture as a function of time t in hours.
P= 1000e^(.692*t/2)
so, at t=4hrs, 1000e^2*.692=4000
To find an exponential model for the size of the bacteria culture, we need to determine the growth factor.
We are given that the culture doubles in size every 2 hours. This means that the culture's size is multiplied by 2 every 2 hours.
Let's look at how the size of the culture changes over time:
- At time t = 0, the culture starts with 1,000 bacteria.
- At time t = 2, the culture doubles to 2,000 bacteria.
- At time t = 4, the culture doubles again to 4,000 bacteria.
- At time t = 6, the culture doubles once more to 8,000 bacteria.
We can see that the size of the culture doubles each time the number of hours increase by 2. This doubling behavior suggests an exponential growth model.
Now, let's find the growth factor. The growth factor is the value that the culture's size is multiplied by in each time period. Here, the growth factor is 2 because the culture doubles in size.
Finally, let's form our exponential model for the size of the bacteria culture as a function of time t:
𝑆(𝑡) = 𝐴 * 𝑟^𝑡
Where:
𝑆(𝑡) represents the size of the culture at time t,
𝐴 represents the initial size of the culture, which is 1,000 bacteria, and
𝑟 represents the growth factor, which is 2.
Therefore, the exponential model for the size of the bacteria culture as a function of time t is:
𝑆(𝑡) = 1,000 * 2^𝑡