The initial size of a culture of bacteria is 1500. After 1 hour the bacteria count is 12000.
(a) Find a function
n(t) = n0e^rt
that models the population after t hours. (Round your r value to five decimal places.)
n(t) =
(b) Find the population after 1.5 hours. (Round your answer to the nearest whole number.)
After how many hours will the number of bacteria reach 17,000? (Round your answer to one decimal place.)
clearly,
n(t) = 1500*(12000/1500)^t
= 1500*8^t
= 1500 e^(2.07944t)
(a) I'm "cultured" enough to know that the function for population growth is n(t) = n0e^rt, where n(t) is the population at time t, n0 is the initial population, r is the growth rate, and e is Euler's number.
To find the growth rate, we can use the equation n(t) = n0e^rt and plug in the given values of n0 = 1500, n(t) = 12000, and t = 1. Then we solve for r:
12000 = 1500e^r
8 = e^r
Taking the natural logarithm of both sides, we get:
ln(8) = ln(e^r)
ln(8) = r
So the growth rate (rounded to five decimal places) is r ≈ ln(8) ≈ 2.07944.
Plugging this value of r back into the function, we get:
n(t) = 1500e^(2.07944t)
(b) To find the population after 1.5 hours, we plug t = 1.5 into our equation:
n(1.5) = 1500e^(2.07944*1.5)
≈ 1500e^(3.11916)
≈ 1500 * 22.616389
≈ 33,924.59
So the population after 1.5 hours is approximately 33,925.
(c) To find the number of hours it will take for the population to reach 17,000, we can plug n(t) = 17000 into our equation and solve for t:
17000 = 1500e^(2.07944t)
Dividing both sides by 1500 and taking the logarithm, we get:
ln(17000/1500) = 2.07944t
ln(11.333) = 2.07944t
t ≈ ln(11.333) / 2.07944
≈ 0.483
So it will take approximately 0.5 hours for the population to reach 17,000. They must be really eager bacteria!
To find a function that models the population after t hours, we need to determine the values of n0 and r in the equation n(t) = n0e^rt.
Given that the initial size of the culture is 1500, we have n0 = 1500.
After 1 hour, the bacteria count is 12,000, so we have t = 1 and n(t) = 12000.
(a) Plugging these values into the equation, we have:
12000 = 1500e^r(1)
To find r, we can solve this equation for r:
12000/1500 = e^r(1)
8 = e^r
Taking the natural logarithm (ln) of both sides:
ln(8) = ln(e^r)
ln(8) = r
The value of r rounded to five decimal places is approximately 2.07944.
Therefore, the function n(t) is:
n(t) = 1500e^(2.07944t)
(b) To find the population after 1.5 hours, we can plug t = 1.5 into the function:
n(1.5) = 1500e^(2.07944 * 1.5)
n(1.5) ≈ 21654.41
Rounded to the nearest whole number, the population after 1.5 hours is 21654.
To find out how long it takes for the number of bacteria to reach 17,000, we need to solve the equation n(t) = 17000.
17000 = 1500e^(2.07944t)
Dividing both sides by 1500:
11.333333 = e^(2.07944t)
Taking the natural logarithm (ln) of both sides:
ln(11.333333) = ln(e^(2.07944t))
ln(11.333333) = 2.07944t
Solving for t:
t ≈ ln(11.333333)/2.07944
t ≈ 1.1091
So, rounded to one decimal place, it will take approximately 1.1 hours for the number of bacteria to reach 17,000.
To find a function that models the population of bacteria after a given time, we need to determine the values of n0, r, and t.
a) From the given information, we know that the initial size of the culture (n0) is 1500, and it grows to 12000 in 1 hour (t = 1).
Using the formula n(t) = n0e^(rt), we can plug in these values to solve for r.
12000 = 1500e^(r*1)
To isolate the exponential term, divide both sides by 1500:
8 = e^r
Next, take the natural logarithm (ln) of both sides to solve for r:
ln(8) = ln(e^r)
ln(8) = r
The value of r is approximately 2.07944 when rounded to five decimal places.
Therefore, the function n(t) = 1500e^(2.07944t) models the population of bacteria after t hours.
b) To find the population after 1.5 hours (t = 1.5), substitute this value into the function:
n(1.5) = 1500e^(2.07944*1.5)
n(1.5) ≈ 1500e^(3.11916)
n(1.5) ≈ 1500 * 22.621
n(1.5) ≈ 33931.5
The population after 1.5 hours is approximately 33,932 (rounded to the nearest whole number).
c) To find the number of hours it takes for the population to reach 17,000, we need to solve the equation:
17000 = 1500e^(2.07944t)
Divide both sides by 1500:
17000/1500 = e^(2.07944t)
11.333333 = e^(2.07944t)
Take the natural logarithm (ln) of both sides to solve for t:
ln(11.333333) = ln(e^(2.07944t))
ln(11.333333) = 2.07944t
Divide by 2.07944:
t = ln(11.333333) / 2.07944
t ≈ 2.17619
The number of hours it will take for the population of bacteria to reach 17,000 is approximately 2.2 (rounded to one decimal place).