A biologist grows a culture of bacteria as part of an experiment. At the start of the experiment, there are 75 bacteria in the culture. The biologist observes that the population of bacteria dobules evert 18 minutes. Which of the following equations best models the number, n, of bacteria t hours after the start of the experiment.
a. n=75(2)^t/18
b. n=75(1+t/18)
c. n=75(2)^10t/3
d. n=75(1+(10/3)t)
a would be a perfect answer if t was in minutes. But it is in hours. There are 10/3 eighteen minutes in an hour, and the number of doubling periods is 10t/3
whoops, sorry, did not notice the hours/minutes problem
when t = 0, n = 75 (note 2^0 = 1)
when t = 18, n = 75 * 2^1 (note 18/18 = 1)
when t = 36 , n = 75 * 2^2 (note 36/18 = 2 ) etc
the 10t/3 that bob talks about is in c
Well, I'm no bacteria expert, but I do know a thing or two about math. Let's break down these equations!
a. n=75(2)^t/18
Hmm, this equation seems a bit off. I don't think dividing t by 18 is the way to go here. Bacteria don't care about dividing their time!
b. n=75(1+t/18)
Ah, now this equation seems to be taking into account the time factor. Adding t/18 to 1 seems like a reasonable approach. But let's keep exploring.
c. n=75(2)^10t/3
Well, this equation looks like it has had a bit too much fun with exponents and 10's. While I appreciate the creativity, it might not accurately model the situation.
d. n=75(1+(10/3)t)
Ah, here we go! This equation seems to strike the right balance. It takes into account the time factor with (10/3)t, while also multiplying it by 75 and factoring in the initial 1. Good job, equation!
So, after all that analysis, I would say the correct answer is option d. Keep those bacteria multiplying, biologist!
To determine the equation that best models the number of bacteria, we need to understand the relationship between time and the growth of the bacteria population.
The problem states that the population of bacteria doubles every 18 minutes. However, the equation options are given in terms of hours. We need to convert the time from hours to minutes to match the given information.
Since there are 60 minutes in an hour, we can convert the time from hours to minutes by multiplying by 60.
So, t hours is equal to 60t minutes.
After converting the time from hours to minutes, we can now determine the correct equation.
The correct equation should follow the exponential growth formula:
n = initial population * growth factor^time
In this case:
Initial population = 75 (the starting number of bacteria)
Growth factor = 2 (the population doubles)
Time = t (in minutes)
Now let's evaluate each option:
a. n = 75(2)^(t/18) - Incorrect
This equation uses the correct initial population (75) but has the growth factor raised to the power of t divided by 18. Since t is in minutes, this equation doesn't match the given information where bacteria double every 18 minutes.
b. n = 75(1 + t/18) - Incorrect
This equation has the correct initial population (75), but it uses a linear growth approach instead of exponential growth. In linear growth, the population increases by a constant rate, but in this case, bacteria double every 18 minutes, so linear growth doesn't apply.
c. n = 75(2)^(10t/3) - Incorrect
Similar to option a, this equation applies the correct initial population (75) but raises the growth factor to the power of 10t/3. Again, this doesn't match the given information where bacteria double every 18 minutes.
d. n = 75(1 + (10/3)t) - Correct
This equation has the correct initial population (75) and represents exponential growth with a growth factor of 2. Since t is in minutes, the 10/3 term inside the parentheses is used to convert the minutes to the appropriate scale.
Therefore, the correct equation is n = 75(1 + (10/3)t), option d.