population growth model. Can anybody please help me out in trying to solve this problem? It's my homework and I don't seem to understand what I am getting.
The count in a culture of bacteria was 400 after 2 hours and 25,600 after 6 hours.
(a) What is the relative rate of growth of the bacteria population? Express your answer as a percentage. (Round your answer to the nearest whole number.)
= ? %
(b) What was the initial size of the culture? (Round your answer to the nearest whole number.)
=# ? bacteria
(c) Find a function that models the number of bacteria n(t) after t hours. (Round your r value to two decimal places.)
n(t) =
(d) Find the number of bacteria after 4.5 hours. (Round your answer to the nearest hundred.)
=#? bacteria
(e) After how many hours will the number of bacteria reach 50,000? (Round your answer to two decimal places.)
=#? hr
general equation:
number = a e^(kt) , where a is the initial amount, t is the time in hours and k is a constant.
case1: when t = 2, number = 400
400 = a e^(2k)
case2: when t = 6 , number = 25600
25600 = a e^(6k)
divide the 2nd equation by the 1st
64 = e^(4k)
4k = ln64
k = ln64/4 = 1.039721
back in 1st:
400 = a(e^(2(1.039721))
400 = a(8)
a = 50
b) 50
c) see above
d) number = 50 e^(4.5(1.039721) = 9050.9..
= 9100 to the nearest 100
e) 50000 = 50 e^(1.039721 t)
1000 = e^(1.039721t)
ln 1000 = 1.039721t
t = appr 6.64 hours
or
looking at the "exact" value of a = 50 , I would guess that this is "doubling" type of question, and we should find the doubling time "d"
let number = a(2)^(t/d)
400 = a 2^(2/d)
25600 = a 2^(6/d)
divide as before, ...
64 = 2^(4/d)
2^6 = 2^(4/d)
4/d = 6
d = 2/3
in 1st:
400 = a 2^(2/(2/3))
400 = a 2^(3)
a = 50 ---> same as before
so number = 50 2^(t/(2/3))
n(t) = 50 2^(3t/2)
I will do the last one using this new equation
50000 = 50 2^(3t/2)
1000 = 2^(3t/2)
log both sides
log 1000 = (3t/2) log2
3/log2 = 3t/2
3t = 6/log2
t = 2/log2 = 6.64 , just as before
(a) The relative rate of growth can be determined by finding the change in population over the time interval.
Change in population = Final population - Initial population
= 25,600 - 400
= 25,200
Relative rate of growth = (Change in population / Initial population) * 100
= (25,200 / 400) * 100
= 6300%
So, the relative rate of growth of the bacteria population is 6300%.
(b) To find the initial size of the culture, we can use the initial population given in the problem, which is 400.
Therefore, the initial size of the culture is 400 bacteria.
(c) To find a function that models the number of bacteria n(t) after t hours, we can use the exponential growth model.
The general form of the exponential growth model is: n(t) = n0 * e^(rt), where n(t) is the number of bacteria after t hours, n0 is the initial number of bacteria, e is the base of the natural logarithm (approximately 2.71828), and r is the relative rate of growth (in decimal form).
Using the information given, we have:
n(t) = n0 * e^(rt)
25,600 = 400 * e^(r * 6)
64 = e^(6r) (divided both sides by 400)
Taking the natural logarithm (ln) of both sides, we get:
ln(64) = ln(e^(6r))
ln(64) = 6r * ln(e)
ln(64) = 6r (since ln(e) = 1)
Simplifying, we have:
6r = ln(64)
r = ln(64) / 6
Using a calculator, we find r ≈ 0.274.
Thus, the function that models the number of bacteria is n(t) = 400 * e^(0.274t).
(d) To find the number of bacteria after 4.5 hours, we substitute t = 4.5 into the function we found in part (c):
n(4.5) ≈ 400 * e^(0.274 * 4.5)
≈ 400 * e^1.233
≈ 400 * 3.435
Rounding to the nearest hundred, we get the number of bacteria after 4.5 hours ≈ 1374 bacteria.
(e) To find the number of hours it takes for the number of bacteria to reach 50,000, we set n(t) = 50,000 in the function we found in part (c):
50,000 = 400 * e^(0.274t)
Dividing by 400, we get:
e^(0.274t) = 125
Taking the natural logarithm of both sides, we have:
0.274t = ln(125)
Solving for t, we find:
t ≈ ln(125) / 0.274
Using a calculator, we get t ≈ 4.76 hours (rounded to two decimal places).
Therefore, it will take approximately 4.76 hours for the number of bacteria to reach 50,000.
To solve this problem, we can use the exponential population growth model. The general form of this model is:
n(t) = n₀ * e^(r*t)
Where:
- n(t) represents the population size at time t
- n₀ represents the initial population size
- r represents the relative growth rate
- e is the base of the natural logarithm (approximately 2.71828)
- t represents the time in hours
Let's solve each part of the problem step-by-step:
(a) What is the relative rate of growth of the bacteria population? Express your answer as a percentage.
To find the relative rate of growth, we need to find the value of r in the exponential growth model. We can use the information given in the problem:
n(2) = 400 and n(6) = 25600
Plugging these values into the model, we get:
400 = n₀ * e^(2*r)
25600 = n₀ * e^(6*r)
Dividing the second equation by the first equation, we get:
25600/400 = e^(6*r)/e^(2*r)
This simplifies to:
64 = e^(4*r)
Taking the natural logarithm of both sides, we get:
ln(64) = 4*r
Solving for r, we find:
r = ln(64)/4 ≈ 0.42
To express the relative rate of growth as a percentage, we multiply it by 100:
Relative rate = 0.42 * 100 ≈ 42%
Therefore, the relative rate of growth of the bacteria population is approximately 42%.
(b) What was the initial size of the culture?
To find the initial size (n₀), we can use the formula we derived in part (a):
400 = n₀ * e^(2*0.42)
Dividing both sides by e^0.84, we get:
400 / e^0.84 = n₀
Using a calculator to evaluate e^0.84, we find:
400 / 2.318 ≈ 172
Therefore, the initial size of the culture was approximately 172 bacteria.
(c) Find a function that models the number of bacteria n(t) after t hours.
Based on the exponential growth model, we can now write the function:
n(t) = 172 * e^(0.42*t)
(d) Find the number of bacteria after 4.5 hours.
To find the number of bacteria after 4.5 hours, we can substitute t = 4.5 into the function we derived in part (c):
n(4.5) ≈ 172 * e^(0.42*4.5)
Using a calculator to evaluate e^(0.42*4.5), we find:
n(4.5) ≈ 172 * e^1.89 ≈ 172 * 6.626 ≈ 1139
Therefore, the number of bacteria after 4.5 hours is approximately 1139.
(e) After how many hours will the number of bacteria reach 50,000?
To find the number of hours it takes for the population to reach 50,000, we can solve the following equation:
50000 = 172 * e^(0.42*t)
Dividing both sides by 172, we get:
50000 / 172 = e^(0.42*t)
Taking the natural logarithm of both sides, we have:
ln(50000/172) = 0.42*t
Solving for t, we find:
t = ln(50000/172) / 0.42 ≈ 10.52
Therefore, it will take approximately 10.52 hours for the number of bacteria to reach 50,000.
To solve this problem, we can use the concept of exponential growth. The general formula for exponential growth is given by:
n(t) = n0 * e^(rt)
Where:
n(t) is the population size at time t
n0 is the initial population size
e is the mathematical constant approximately equal to 2.71828
r is the relative growth rate
t is the time in hours
(a) To find the relative rate of growth, we need to calculate the value of r. We have two data points: n(2) = 400 and n(6) = 25,600. Plugging these values into the exponential growth formula gives us:
400 = n0 * e^(2r) ----(1)
25,600 = n0 * e^(6r) ----(2)
Dividing equation (2) by equation (1), we get:
64 = e^(6r - 2r)
64 = e^(4r)
To solve for r, we take the natural logarithm of both sides:
ln(64) = ln(e^(4r))
Using the property of logarithms, we can bring down the exponent:
ln(64) = 4r * ln(e)
Since ln(e) is equal to 1, the equation simplifies to:
ln(64) = 4r
Solving for r:
r = ln(64) / 4 ≈ 0.871
To express the relative rate of growth as a percentage, we multiply r by 100:
Relative rate of growth ≈ 0.871 * 100 ≈ 87.1%
Therefore, the relative rate of growth of the bacteria population is approximately 87.1%.
(b) To find the initial size of the culture (n0), we can plug the values of r and one of the data points (400) into the exponential growth formula:
400 = n0 * e^(2 * 0.871)
Solving for n0:
n0 = 400 / e^(1.742) ≈ 160.80
Rounding to the nearest whole number, the initial size of the culture is approximately 161 bacteria.
(c) The function that models the number of bacteria (n(t)) after time t hours is given by:
n(t) = 161 * e^(0.871t)
(d) To find the number of bacteria after 4.5 hours, we can plug t = 4.5 into the function:
n(4.5) = 161 * e^(0.871 * 4.5)
Calculating this value gives:
n(4.5) ≈ 161 * e^(3.919) ≈ 161 * 50.551 ≈ 8,147
Rounding to the nearest hundred, the number of bacteria after 4.5 hours is approximately 8,100 bacteria.
(e) To find the number of hours it will take for the population to reach 50,000 bacteria, we can rearrange the formula to solve for t:
50,000 = 161 * e^(0.871t)
Divide both sides by 161:
50,000 / 161 = e^(0.871t)
Taking the natural logarithm of both sides:
ln(50,000 / 161) = 0.871t
Solving for t:
t = ln(50,000 / 161) / 0.871 ≈ 6.47
Rounding to two decimal places, the number of hours it will take to reach 50,000 bacteria is approximately 6.47 hours.