Researchers use formula f(t) = Pe^rt to model the population of fruit flies growing exponentially after t days.
Measurements show that f(0) = 30 and
f(5) = 1620
a) Find a function of the form
f(t) = Pe^rt that models the fruit fly population after t days.
b) When the function f is expressed in the f(x) = Pa^x, the constant a is the growth factor. Find the growth factor.
c) When will there be 18,000 fruit flies?
f(0) = 30 , so
30 = P(e^0)
P = 30
f(5) = 1620
1620 = 30 e^(5r)
54 = e^(5r(
ln 54 = ln e^(5r)
ln54 = 5r
r = ln54/5 = .7978
f(t) = 30 e^(.7978t)
replace f(t) with 18000 and solve for t
a) To find a function that models the fruit fly population after t days, we use the given information f(0) = 30 and f(5) = 1620.
First, we know that f(0) = 30, which means that when t = 0, the population is 30. This gives us the initial condition P = 30.
Next, we have f(5) = 1620, which means that when t = 5, the population is 1620. Plugging these values into the formula, we get:
1620 = 30 * e^(5r)
Dividing both sides by 30:
54 = e^(5r)
Taking the natural logarithm (ln) of both sides:
ln(54) = ln(e^(5r))
ln(54) = 5r
Now we can solve for r by dividing both sides by 5:
r = ln(54) / 5
Therefore, the function that models the fruit fly population after t days is:
f(t) = 30 * e^((ln(54) / 5) * t)
b) In the expression f(x) = Pa^x, the constant a is the growth factor. To find the growth factor, we can rewrite our function from part a:
f(t) = 30 * e^((ln(54) / 5) * t)
Comparing this to f(x) = Pa^x, we can see that the growth factor a is e^(ln(54) / 5).
Therefore, the growth factor is e^(ln(54) / 5).
c) To find when there will be 18,000 fruit flies, we need to solve for t in the function.
18,000 = 30 * e^((ln(54) / 5) * t)
We can simplify this equation by dividing both sides by 30:
600 = e^((ln(54) / 5) * t)
Taking the natural logarithm of both sides:
ln(600) = (ln(54) / 5) * t
Now we can solve for t by dividing both sides by (ln(54) / 5):
t = ln(600) / (ln(54) / 5)
Plug this into a calculator, and you will find the value of t when there will be 18,000 fruit flies.