A culture of bacteria triples every 7 minutes. How long will it take a culture originally consisting of š4 bacteria to grow to a population of š00 000 bacteria? Show ALL work and round to one decimal place, if necessary.
I'll help you with the solution. Showing ALL work is your job. Just post the question.
the population after t minutes is 24*3^(t/7)
so, you need to solve
24*3^(t/7) = 100000
3^(t/7) = 12500/3
t/7 log3 = log(12500/3)
Now finish it off
To solve this problem, we can use the exponential growth formula:
N = Nā * (1 + r)^t
Where:
- N is the final population size
- Nā is the initial population size
- r is the growth rate per unit of time
- t is the time in units
In this case, the initial population size (Nā) is 24 bacteria, and we want to find the time it takes for the population to reach 100,000 bacteria (N). The growth rate (r) can be determined from the information given: the population triples every 7 minutes. This means that the growth rate is 3 per 7 minutes, or 3/7.
Substituting the given values into the formula, we have:
100,000 = 24 * (1 + 3/7)^t
To isolate the exponent, we'll divide both sides of the equation by 24:
100,000 / 24 = (1 + 3/7)^t
4,166.67 = (1 + 3/7)^t
Next, we'll take the logarithm of both sides of the equation to solve for t:
log(4,166.67) = log((1 + 3/7)^t)
Using logarithmic properties, we can bring the exponent down:
log(4,166.67) = t * log(1 + 3/7)
Now, we can calculate the value of t by dividing the logarithm of 4,166.67 by the logarithm of (1 + 3/7).