The number of bacteria in a certain population increases according to a continuous exponential growth model, with a growth rate parameter of 3.7%
per hour. How many hours does it take for the size of the sample to double?
Note: This is a continuous exponential growth model.
Do not round any intermediate computations, and round your answer to the nearest hundredth.
Why did the bacteria go to school? To get a little culture!
Now, let's solve the problem. In a continuous exponential growth model, the formula is given by:
N(t) = N_0 * e^(rt),
Where N(t) is the size of the sample at time t, N_0 is the initial size of the sample, e is Euler's number (approximately 2.71828), and r is the growth rate parameter.
In this case, we want to find the time it takes for the size of the sample to double, so N(t) = 2N_0.
Setting N(t) to 2N_0 and solving for t, we have:
2N_0 = N_0 * e^(0.037t).
Dividing both sides by N_0, we get:
2 = e^(0.037t).
Now, take the natural logarithm of both sides:
ln(2) = 0.037t * ln(e).
Since ln(e) = 1, we simplify the equation to:
ln(2) = 0.037t.
Now, solve for t:
t = ln(2) / 0.037.
Using a calculator, we find:
t ≈ 18.72.
So, it takes approximately 18.72 hours for the size of the sample to double. But remember, laughter is the best medicine, not exponential growth!
To find the number of hours it takes for the size of the sample to double, we can use the formula for exponential growth:
A = P * e^(rt)
Where:
A = final size of the sample
P = initial size of the sample
e = Euler's number (approximately 2.71828)
r = growth rate parameter (expressed as a decimal)
t = time in hours
In this case, we know that the growth rate parameter is 3.7% per hour, which is equivalent to 0.037 in decimal form.
We are looking for the value of t when the final size of the sample (A) is double the initial size of the sample (P), so A = 2P.
Substituting these values into the formula:
2P = P * e^(0.037t)
Dividing both sides of the equation by P:
2 = e^(0.037t)
We need to solve for t. Taking the natural logarithm (ln) of both sides of the equation allows us to isolate the exponent:
ln(2) = 0.037t * ln(e)
Since ln(e) = 1, the equation simplifies to:
ln(2) = 0.037t
To solve for t, divide both sides of the equation by 0.037:
t = ln(2) / 0.037
Using a calculator, we find:
t ≈ 18.71
Therefore, it takes approximately 18.71 hours for the size of the sample to double.
To solve this problem, we'll use the continuous exponential growth model equation:
N(t) = N₀ * e^(rt)
Where:
- N(t) is the size of the population at time t
- N₀ is the initial size of the population
- r is the growth rate parameter
- e is Euler's number (approximately 2.71828)
We want to find the time it takes for the population size to double, so we'll set the final population size, N(t), to be twice the initial size, N₀.
2 * N₀ = N₀ * e^(rt)
We can simplify this equation by dividing both sides by N₀:
2 = e^(rt)
To solve for t, we need to take the natural logarithm (ln) of both sides:
ln(2) = ln(e^(rt))
Using the property of logarithms, this simplifies to:
ln(2) = rt * ln(e)
Since ln(e) is equal to 1, we have:
ln(2) = rt
Finally, we can solve for t by dividing both sides by r:
t = ln(2) / r
Now, plugging in the given growth rate parameter r = 0.037 (which is 3.7% expressed as a decimal), we can calculate the time it takes for the population size to double.
t = ln(2) / 0.037 = 18.66666667
Rounding this to the nearest hundredth, we find that it takes approximately 18.67 hours for the size of the sample to double.
the growth after t hours is 1.036^t
so, find t where
1.036^t = 2
t log1.036 = log2
t = log2/log1.036 = 19.5986