The initial size of a culture of bacteria is 1000. After one hour the bacteria count is 4000.
(a) Find a function
n(t) = n0ert
that models the population after t hours.
n(t) =
b)Find the population after 1.5 hours. (Round your answer to the nearest whole number.)
n(1.5) = bacteria
a) To find a function that models the population after t hours, we can use the given information that the initial size of the culture is 1000 and the bacteria count after one hour is 4000.
Using the formula n(t) = n0 * e^(rt), where n0 represents the initial size, r represents the growth rate, and t represents time, we can solve for r:
4000 = 1000 * e^(1 * r)
4 = e^r
Taking the natural logarithm (ln) of both sides of the equation:
ln(4) = ln(e^r)
ln(4) = r
Now we have the value for r, which is ln(4). Plugging this value into the formula, we get:
n(t) = 1000 * e^(ln(4) * t)
b) To find the population after 1.5 hours, we substitute t = 1.5 into the formula:
n(1.5) = 1000 * e^(ln(4) * 1.5)
Calculating this expression, we get:
n(1.5) ≈ 1000 * e^(ln(4) * 1.5) ≈ 1000 * e^(1.386) ≈ 1000 * 4 ≈ 4000
So, the population after 1.5 hours is approximately 4000 bacteria.
(a) To find a function n(t) = n0ert that models the population after t hours, we need to find the values of n0 (initial population size), r (growth rate), and t (time in hours).
Given:
n0 = 1000 (initial size of the culture)
t = 1 (time in hours)
n(t) = 4000 (population after 1 hour)
To find r, we can use the formula:
n(t) = n0ert
Plugging in the known values:
4000 = 1000 * er
Simplifying:
4 = e^r
To solve for r, take the natural logarithm (ln) on both sides:
ln(4) = ln(e^r)
Using the property of logarithms (ln(e^r) = r), we have:
ln(4) = r
Using a calculator, we find that r ≈ 1.386.
Now we can substitute the values of n0, r, and t back into the equation n(t) = n0ert:
n(t) = 1000 * e^(1.386t)
(b) To find the population after 1.5 hours, we substitute t = 1.5 into the equation n(t) = 1000 * e^(1.386t):
n(1.5) = 1000 * e^(1.386 * 1.5)
Using a calculator to evaluate the expression, we find that n(1.5) ≈ 5488 bacteria (rounded to the nearest whole number).
To find a function that models the population after t hours, we can use the exponential growth formula n(t) = n0 * e^(rt).
Given that the initial size of the culture is 1000 (n0 = 1000) and after one hour the bacteria count is 4000, we need to find the value of "r".
The formula becomes: 4000 = 1000 * e^(r * 1)
To isolate "r", divide both sides of the equation by 1000:
4 = e^r
Now, take the natural logarithm (ln) of both sides to solve for "r":
ln(4) = ln(e^r)
ln(4) = r
So, the value of "r" is ln(4).
a) Therefore, the function that models the population after t hours is:
n(t) = 1000 * e^(ln(4) * t)
b) To find the population after 1.5 hours, we substitute t = 1.5 into the function:
n(1.5) = 1000 * e^(ln(4) * 1.5)
Now we can calculate the population using this equation.
n = 1000 e^(kt)
note when t = 0, e^kt = e^0 = 1
so 1000*1 at t =0
4000/1000 = 4 = e^(k*1)
or
e^k = 4
k = ln 4 = 1.386
so
n = 1000 e^ (1.386 t)
it t = 1.5
n = 1000 e^(2.079)
n = 8000