The function f(t)=100,000/1+5000e^(-t) represents the population of an endangered species bird t years after they were introduced to a non-threatening habitat.
a.In what year will there be 70,000 birds?
b.How many birds were intially introduced to the habitat?
c.How many birds are expected in the habitat eleven years after the introduction?
I'm thinking y=y0e^(-t)as the formula.
Looking for advice on set up of calculation, not necessarily answers to all.
a) So, 70000 = 10000/(1 + 5000e^(-t)
1+5000e^-t = 10/7
5000e^-t = 10/7 - 1 = 3/7
e^-t = 3/35000 = .000085714
ln both sides
-t = ln .000085714
-t = -9.36
t = 9.36 years
b) when t=0
f(0) = 100000(1+ e^0) = 100000/2 = 50000
c) plug in t=11, you do the button-pushing
To find the year when there will be 70,000 birds (a), you need to solve for t in the function f(t) = 70,000.
a) So, we have:
70,000 = 100,000 / (1 + 5000e^(-t))
To solve for t, we can start by multiplying both sides of the equation by (1 + 5000e^(-t)) to get rid of the fraction:
70,000(1 + 5000e^(-t)) = 100,000
Now, distribute the 70,000 on the left side:
70,000 + 350,000,000e^(-t) = 100,000
Subtract 70,000 from both sides:
350,000,000e^(-t) = 30,000
Divide both sides by 350,000,000 to isolate e^(-t):
e^(-t) = 30,000 / 350,000,000
Now, take the natural logarithm (ln) of both sides to solve for t:
ln(e^(-t)) = ln(30,000 / 350,000,000)
Using the property of logarithms that ln(e^x) = x, we can simplify further:
-t = ln(30,000 / 350,000,000)
Multiply both sides by -1 to solve for t:
t = -ln(30,000 / 350,000,000)
Now, plug this value of t into (current year - t) to find the year with 70,000 birds.
b) To find how many birds were initially introduced to the habitat, we need to find the population when t = 0. So, we can simply plug in t = 0 into the function f(t):
f(0) = 100,000 / (1 + 5000e^(-0))
Simplifying, we get:
f(0) = 100,000 / (1 + 5000)
f(0) = 100,000 / 5,001
So, the initial population is 100,000 / 5,001 birds.
c) To find the expected population eleven years after the introduction (t = 11), we can plug t = 11 into the function f(t):
f(11) = 100,000 / (1 + 5000e^(-11))