Bacteria grow in a nutrient solution at a rate proportional to the amount present. Initially, there are 250 strands of the bacteria in the solution which grows to 800 strands after seven hours. Find the time needed for the bacteria to grow to 1600 strands.
Given a starting population of N0, our general expression for exponential growth is N=N0ert
where r is a positive constant. Substituting for the given information, we can find r as follows:
800=250er(7)
3.20=e7r
r=17ln3.2=0.166164
a) A general expression for the number of stands in the culture at any time is N=250e0.166164t
.
b) The time needed for the bacteria to grow to 1,600 strands is therefore:
1600=250e(0.166164)t
6.4=e0.166164t
t=10.166164ln6.4=11.1715
Therefore, it takes 11.17 hours for the population to grow to 1,600 strands.
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i dont know the answers please help me asap
Oh, bacterial growth, eh? Let me just put on my lab coat and goggles. Now, the growth of bacteria follows an exponential function with time. We can use the formula:
N(t) = N0 * e^(kt)
Where N(t) is the number of strands at time t, N0 is the initial number of strands, e is the mathematical constant approximately equal to 2.71828, and k is the rate constant.
Given that initially there are 250 strands (N0 = 250) and after seven hours it grows to 800 strands (N(t) = 800), we can plug these values in to solve for k:
800 = 250 * e^(7k)
Divide both sides by 250:
8/25 = e^(7k)
Take the natural log of both sides:
ln(8/25) = 7k
Now we can solve for k:
k = ln(8/25) / 7
Now that we have our rate constant, we can find the time needed for the bacteria to grow to 1600 strands (N(t) = 1600). Let's plug the values into our formula:
1600 = 250 * e^((ln(8/25)/7) * t)
Divide both sides by 250:
16/25 = e^((ln(8/25)/7) * t)
Take the natural log of both sides:
ln(16/25) = (ln(8/25)/7) * t
Finally, solve for t:
t = (7 * ln(16/25)) / ln(8/25)
And there you have it, the amount of time needed for the bacteria to grow to 1600 strands. Just remember, they're growing so fast they'll have to start their own bacterial band soon!
To solve this problem, we can use exponential growth.
Let's define the initial amount of bacteria as N₀ and the final amount of bacteria as N.
Given that bacteria grow at a rate proportional to the amount present, we can write the equation:
dN/dt = kN
where dN/dt is the rate of change of the amount of bacteria with respect to time, and k is the proportionality constant.
Now, let's find the value of k. We have the initial condition:
N(t=0) = 250
Substituting these values into our equation, we get:
dN/dt = k * 250
To find the value of k, we can use the information given that the bacteria grows to 800 strands after seven hours:
N(t=7) = 800
Substituting these values into our equation, we get:
dN/dt = k * 800
Now, we have two equations:
dN/dt = k * 250
dN/dt = k * 800
Since the left sides are the same, we can equate the right sides:
k * 250 = k * 800
Simplifying, we find:
250 = 800
Dividing both sides by 250, we get:
k = 3.2
Now, we have the following differential equation:
dN/dt = 3.2N
To solve this differential equation, we can use the separation of variables technique. We separate the variables and integrate both sides:
1/N dN = 3.2 dt
Integrating both sides, we get:
ln|N| = 3.2t + C
where C is the constant of integration.
Next, we exponentiate both sides:
|N| = e^(3.2t + C)
Since N is the number of bacteria, it cannot be negative. Therefore, we can remove the absolute value:
N = e^(3.2t + C)
Now, we can substitute the initial condition N(t=0) = 250 into the equation to find the constant of integration C:
250 = e^(3.2(0) + C)
250 = e^C
Taking the natural logarithm of both sides, we get:
ln(250) = C
So, the equation becomes:
N = e^(3.2t + ln(250))
Now, let's find the time needed for the bacteria to grow to 1600 strands. We can substitute N = 1600 into our equation and solve for t:
1600 = e^(3.2t + ln(250))
Dividing both sides by 250, we get:
1600/250 = e^(3.2t)
Simplifying, we find:
6.4 = e^(3.2t)
Taking the natural logarithm of both sides, we get:
ln(6.4) = 3.2t
Dividing both sides by 3.2, we find:
t = ln(6.4)/3.2
Calculating this value, we find:
t ≈ 0.523
Therefore, it takes approximately 0.523 hours (or about 31.4 minutes) for the bacteria to grow to 1600 strands.