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.
f(t) =
1
To find an exponential model for the size of the bacteria culture as a function of time, we need to determine the growth rate.
Given that the culture doubles in size every 2 hours, we can use this information to find the growth rate by taking the formula for exponential growth:
f(t) = P * (1 + r)^t
Where:
- f(t) represents the size of the culture at time t
- P represents the initial size of the culture
- r represents the growth rate
- t represents the time in hours
In this case, the initial size of the culture (P) is 1,000 bacteria. We need to find the growth rate (r) per hour.
Since the culture doubles in size every 2 hours, we can express this as (1 + r)^2 = 2, because after 2 hours the size doubles. Solving this equation for r:
(1 + r)^2 = 2
Take the square root of both sides:
1 + r = sqrt(2)
Subtract 1 from both sides:
r = sqrt(2) - 1
Now that we have the growth rate (r), we can substitute it into the formula for exponential growth:
f(t) = 1,000 * (1 + (sqrt(2) - 1))^t
Simplifying this equation would give us the exponential model for the size of the bacteria culture as a function of time (t).