Rate of change proportional to size:
P(t)= P(i)e^kt, where P is population, t is time, k is a constant and P(i) is the initial p or "p naught"; sorry, I don't know the typed notation for subscript. By the way, p(i) is given as 100.
The problems are
a.) "Find the expression after t hours"
b.) "Find the number of bacteria after t hours."
The answer to a is(via back of the book) is: 100(4.2)^t
I'm not sure exactly how to get there though I believe it has to do with taking natural logarithms and solving for e^k but since p is a function of t I'm not sure how to differentiate "k" (treat it as a constant or differentiate implicitly). If you could show the step by step process I'd appreciate it.
I got to: p(t)= 100(e^t+e^k) and stalled out.
Thanks in advance!
Without some specific numerical information on growth rate or the value of k, I don't see how that back-of-the-book equation can be derived. Is t in seconds in the original equation?
I also don't see the difference between "the population" and "the number of bacteria"
To find the expression after t hours, we can substitute the given values into the equation P(t) = P(i)e^kt.
Given: P(i) = 100 and t = t
Substituting these values, we have P(t) = 100e^kt.
To find the number of bacteria after t hours, we need to evaluate the expression using a specific value for t. Let's call it t = t1.
Substituting t = t1 into the expression, we have P(t1) =100e^k(t1).
Now, let's solve the problem step by step:
a.) Find the expression after t hours:
To simplify the expression, let's write it as: P(t) = 100e^(kt).
b.) Find the number of bacteria after t hours:
Substituting t = t1, we have P(t1) = 100e^(kt1).
Please note that in your question, you mentioned the answer to be 100(4.2)^t. However, based on the provided equation, the correct answer should be P(t) = 100e^kt.
Differentiating the equation P(t) = 100e^(kt) with respect to t will help find the rate of change. Let's find the derivative:
dP(t)/dt = 100ke^(kt).
This derivative gives us the rate of change of the population with respect to time.
I hope this helps! Let me know if you have any further questions.