Answer :
The time in minutes when the petri dish contain 10,000 bacteria is 159.36 minutes.
What is defined as the rate of increase?
- The momentum of a variable is represented by the rate of change, which is used to mathematically define the percentage change in value above a defined period of time.
- The rate of change equation describes the relationship between how each quantity shifts in relation to how another quantity changes.
For the given question;
The time of doubling of number of bacteria in a petri dish = 38.4 minutes.
Present number of bacteria = 562.
Final number of bacteria = 10,000.
Let the 'x' be the current number of bacteria.
Let 'n' be the time of doubling.
Then,
2ⁿx = Final bacteria
Put the values;
2ⁿ×562 = 10, 000
2ⁿ = 10,000/562
2ⁿ = 17.79
Taking natural log both side;
㏑2ⁿ = ㏑17.79
Using the power law of log.
n ㏑2 = ㏑17.79
n = ㏑17.79/㏑2
Solve using calculator;
n = 4.15
The doubling rate was every 38.4 minutes.
Thus, for n = 4.15.
Thus, time = 4.15× 38.4 minutes.
Time = 159.36 minutes.
Therefore, the time after which the number of bacteria in the petri dish will be 10,000 is 159.36 minutes.
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Final answer:
Using the exponential growth formula, we calculated that it would take approximately 114.57 minutes for the bacteria to grow from 562 to 10,000 in a petri dish with a doubling time of 38.4 minutes.
Explanation:
To determine the number of minutes it will take for the number of bacteria to increase from 562 to 10,000 with a doubling time of every 38.4 minutes, we can use the formula for exponential growth:
N = N_0 × 2^(t/T)
where N is the final number of bacteria, N_0 is the initial number of bacteria, t is the total time in minutes, and T is the doubling time in minutes.
First, let's isolate t in the formula:
t = T × (log(N/N_0) / log(2))
We know that N = 10,000, N_0 = 562, and T = 38.4 minutes. Plugging these values into the formula, we obtain:
t = 38.4 × (log(10,000/562) / log(2))
After performing the calculations, we find that t is approximately 114.57 minutes. Therefore, it will take roughly 114.57 minutes for the number of bacteria to grow from 562 to 10,000.
