Final answer:
To find the initial population, we can use the formula for exponential growth. The initial population is approximately 74.85 bacteria. The exponential growth model for the bacteria population is P(t) = 74.85 * e^(ln(2.8)/2 * t).
Step-by-step explanation:
To find the initial population, we can use the formula for exponential growth: P(t) = P(0) * e^(kt), where P(t) is the population at time t, P(0) is the initial population, k is the growth rate, and e is the base of the natural logarithm. We can use the given information to set up two equations:
- P(2) = P(0) * e^(2k)
- = 125
- P(4) = P(0) * e^(4k)
- = 350
e^(4k - 2k) = 350/125
= 2.8
Simplifying, we have:
e^(2k) = 2.8
Taking the natural logarithm of both sides, we get:
2k = ln(2.8)
k = ln(2.8)/2
Substituting the value of k back into one of the original equations, we can solve for P(0):
P(0) = 125 / e^(2 * ln(2.8)/2)
Simplifying further:
P(0) = 125 / (e^(ln(2.8))^(1/2))
P(0) = 125 / 2.8^(1/2)
P(0) = 125 / 1.67
P(0) = 74.85
Therefore, the initial population is approximately 74.85 bacteria.
b) To write an exponential growth model for the bacteria population, we can use the formula P(t) = P(0) * e^(kt), where P(t) is the population at time t, P(0) is the initial population, k is the growth rate, and e is the base of the natural logarithm.
Substituting the initial population and growth rate we found earlier, the exponential growth model is
P(t) = 74.85 * e^(ln(2.8)/2 * t).
c) To use the model, you can plug in the desired time t into the exponential growth model and calculate the population P(t) at that time.