After 3 hours the number of bacteria in a culture is observed to have doubled. Find the time that it takes the culture to grow to 30 times its initial size.
A = Ai e^kt
2 = e^3k
3 k = ln 2
k = .231
30 = e^(.231 t)
ln 30 = .231 t
t = 14.7 hours
To find the time it takes for the culture to grow to 30 times its initial size, we can use the concept of exponential growth.
Let's denote "t" as the time it takes for the culture to grow to 30 times its initial size.
Given that the number of bacteria doubles every 3 hours, we can write the growth equation as:
2^((t/3)) = 30
To solve for "t", we need to isolate the variable.
Taking the logarithm of both sides of the equation, using any base (e.g., natural logarithm):
ln(2^((t/3))) = ln(30)
Using the property of logarithms, we can bring down the exponent:
(t/3) ln(2) = ln(30)
Dividing both sides of the equation by ln(2) gives:
t/3 = ln(30) / ln(2)
Finally, multiplying both sides of the equation by 3 gives:
t = 3 * (ln(30) / ln(2))
Using a calculator, we can evaluate the right-hand side of the equation to find the value of "t".
To find the time it takes for the culture to grow to 30 times its initial size, we can use the concept of exponential growth.
Let's start by understanding the doubling time. Since the number of bacteria in the culture doubles after 3 hours, we can consider this as one "doubling period." This means that for each doubling period, the size of the culture multiplies by a factor of 2.
Now, since the culture needs to grow to 30 times its initial size, we need to determine how many doubling periods are required.
Let's assume that it takes "t" hours for the culture to reach 30 times its initial size. In that case, the growth factor can be represented as 2^t. We know that the culture needs to grow to 30 times its initial size, so we have:
2^t = 30.
To solve for "t" (the number of doubling periods), we need to take the logarithm (base 2) of both sides of the equation:
log2(2^t) = log2(30),
t = log2(30).
Using a calculator or logarithm table, we can find that log2(30) is approximately 4.9.
Therefore, it takes approximately 4.9 doubling periods (or 4.9 * 3 = 14.7 hours) for the culture to grow to 30 times its initial size.