Put the function P= 35(1.3)^t in the form P0ekt When written in this form, you have:
K=
P0=
at t=0, P=35, so Po is 35
P=35 e^kt
P=35(1.3)^t
dividing equations..
1=e^kt/(1.3)^t
or
1.3^t=e^kt
taking ln of each side
t*ln(1.3)=kt
k=ln(1.3)
K = 1.3
P0 = 35
To write the function P = 35(1.3)^t in the form P0e^kt, we need to identify the values of P0 and k.
P0 represents the initial value of P when t = 0.
k represents the growth rate or decay rate.
To find P0 and k, we need to rewrite the given function in the form P = P0e^kt.
Given function: P = 35(1.3)^t
To start, let's write the given equation using the exponential function notation:
P = P0e^kt
Substituting the given values, we have:
35(1.3)^t = P0e^(kt)
Now, we can compare the given equation with the standard form and find the values of P0 and k.
Comparing the exponents, we get:
1.3 = e^k
To solve for k, we will take the natural logarithm (ln) of both sides:
ln(1.3) = ln(e^k)
Using the property that ln(e) = 1, we have:
ln(1.3) = k
Therefore, k = ln(1.3)
Now, to find P0, we will substitute the value of k back into the original equation:
35(1.3)^t = P0e^kt
Since we know k, we can rewrite the equation as:
35(1.3)^t = P0e^(ln(1.3)t)
Simplifying further:
35(1.3)^t = P0(1.3)^t
Comparing the coefficients of (1.3)^t, we get:
P0 = 35
Therefore, when written in the form P0e^kt, the given function P = 35(1.3)^t has:
- k = ln(1.3)
- P0 = 35
To write the function P= 35(1.3)^t in the form P0ekt, we need to find the values of P0 and k.
In the given function, P0 represents the initial value of P when t = 0. To determine P0, substitute t = 0 into the function:
P = 35(1.3)^t
P0 = 35(1.3)^0
P0 = 35(1)
P0 = 35
So, P0 = 35.
Next, let's find the value of k. To do this, we need to rewrite the given function in the form P = P0ekt and isolate k.
P = 35(1.3)^t
Divide both sides by P0:
P / P0 = (35(1.3)^t) / P0
Simplify the right side:
P / P0 = (1.3)^t
Take the natural logarithm (ln) of both sides:
ln(P / P0) = ln((1.3)^t)
Using the logarithmic property ln(a^b) = b * ln(a), rewrite the equation as:
ln(P / P0) = t * ln(1.3)
Finally, solve for k by isolating it:
k = ln(1.3)
Therefore, the final form of the given function is:
P = 35e^(ln(1.3)t)
So, K = ln(1.3) and P0 = 35.