4. BIOLOGY LOGISTICS GROWTH FUNCTION The population of a...

4. BIOLOGY LOGISTICS GROWTH FUNCTION The population of a bacterial culture is modeled by the logistic growth function $y = frac{925}{1 + e^{-0.3t}}$ where $t$ is the time in days and $y$ is the culture weight (in grams). (a) Find the weight of the culture after 0 hours, 1 hours, and 10 hours. (b) What is the limit of the model as $t$ increases without bound? Explain your answer.

Transcript

00:01 In this problem, we're told that the population of a bacterial culture is modeled by the given logistic growth function.

00:07 So we have our equation here.

00:08 And it says that t is representing the time in days, and y is the culture weight in terms of grams.

00:14 So in part a, we want to find the weight of the culture after 0 hours, after 1 hour, and after 10 hours.

00:21 Now, keep in mind it says that t is in terms of days.

00:25 So if we've gone 0 hours, we've gone 0 days.

00:27 So for the first one, for 0 hours, we're going to substitute in 0 for t.

00:32 So we're going to have y equals 925, all divided by 1 plus our constant e, raised to the negative 0 .3 times 0 power.

00:42 So now we can type this into your graphing calculator all at once, as long as you have a set of parentheses around your denominator.

00:48 So i'm going to do 925 divided by, open parentheses, 1 plus our constant e, raised to the negative 0 .3 times 0 power.

00:58 And then i'm going to make sure i close my parentheses afterwards.

01:01 So now, when i put that in, i'll get the weight.

01:04 So that means our initial weight was 462 .5 grams.

01:09 Okay, perfect.

01:10 So next, we need to find the amount of grams after 1 hour.

01:14 Now, again, keep in mind your value for t is in terms of days.

01:20 So 1 hour is 1 24th of a day.

01:23 So that's actually going to be our t value.

01:25 So now we're going to do the same thing, except this time we're plugging in 1 over 24 in place of t.

01:31 So i'm going to have y equals 925 divided by, and then again, in parentheses, 1 plus e raised to the negative 0 .3 times 1 over 24 power.

01:45 And then i'm going to make sure i close that other set of parentheses.

01:48 So we're going to do 925 divided by, open parentheses, 1 plus our constant e, raised to the negative 0 .3 times 1 divided by 24 power.

02:00 And again, we're going to close that last set of parentheses.

02:03 So you'll find that after 1 hour, the weight would be 465 point, and we're going to see, you're told what to round to here...

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