Put the function P=40(39)t in the form P0e^kt. When written in this form, you have:
k= ?
P0= ?
e^ln39 = 39
P0 =40
k = ln39
To put the function P=40(39)t in the form P0e^kt, we need to rewrite it using the properties of exponential functions.
First, let's break down the given function:
P = 40(39)t
To put it in the form P0e^kt, we need to isolate the base, which is e, and rewrite the equation. It's important to note that e is a mathematical constant approximately equal to 2.71828.
Let's start by rewriting the function:
P = 40(39)t
Next, we can rewrite the function using exponential notation:
P = 40 * (e^ln(39))^t
Since ln(39) is a constant, we can replace it with another constant, let's say k, to simplify the notation:
k = ln(39)
Now we can rewrite the function using the constant k:
P = 40 * (e^k)^t
Now, let's rewrite it in the desired form:
P = P0 * e^kt
Comparing the two forms, we can see that:
P0 = 40 (the initial population or value)
k = ln(39) (the constant in the exponent)
So, when written in the form P0e^kt, we have:
P0 = 40
k = ln(39)