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!
e^kt=(e^k)^t so apparently e^k=100
To find the expression for P(t) after t hours, we need to substitute the given values into the equation P(t) = P(i)e^(kt).
Given:
P(i) = 100 (initial population)
P(t) = ? (population after t hours)
k = ? (constant)
Step 1: Substitute the value of P(i) into the equation.
P(t) = 100e^(kt)
Step 2: We need to find the value of k. In order to do this, we need additional information or a specific value for P(t) or P'(t).
Assuming we have more information or a specific value for P(t) or P'(t), we can then use it to determine the value of k and solve for P(t).
For example, if you are given that P(0) = 100, then you can substitute t = 0 into the equation:
100 = 100e^(0*k)
Simplifying, we find that 1 = e^(0) = 1.
This equation does not provide any information about the value of k. Hence, we need additional information or a specific value to proceed further.
Without further information or a specific value, it is not possible to determine the exact expression for P(t) after t hours or find the value of k in the given equation.