Can someone check my math?

[Lorizael]

As I mentioned earlier, I've been reteaching myself Calculus recently. At the moment, I'm trying to figure out how to solve a word problem of my own devising. It goes like this:

1 bacteria consumes 1 unit of food per minute. A colony of this bacteria doubles in size every 60 minutes. If there are 1000 units of food and you start off with 1 bacteria, how long do the bacteria have before the food runs out?

So the formula for the population would be p(t) = e^ln(2)t/60, right? And might the integral of that be 60/ln(2) * e^ln(2)t/60? And then I would take the integral from 0 to x, set that equation equal to 1000, and solve for x, yes? And would my answer be something along the lines of ~219 minutes?

[Lul Thyme]

As I mentioned earlier, I've been reteaching myself Calculus recently. At the moment, I'm trying to figure out how to solve a word problem of my own devising. It goes like this:

1 bacteria consumes 1 unit of food per minute. A colony of this bacteria doubles in size every 60 minutes. If there are 1000 units of food and you start off with 1 bacteria, how long do the bacteria have before the food runs out?

So the formula for the population would be p(t) = e^ln(2)t/60, right? And might the integral of that be 60/ln(2) * e^ln(2)t/60? And then I would take the integral from 0 to x, set that equation equal to 1000, and solve for x, yes? And would my answer be something along the lines of ~219 minutes?

With a strict reading of the problem, I would argue that you made two mistakes:
-the number of bacteria should be a discrete variable,
-it is only stated that the number doubles every hour, not that you have constant relative growth.

The solution would then be something like :
there is one bacteria during the first hour (eating a total of 60 units), two during the second hour (eating a total of 120), 4 during the third (eating a total of 240), 8 during the fourth (eating a total of 480). There are 100 units left after that and 16 bacteria, good enough for 6.25 minutes, for a total of 246.25 minutes.

On one hand, this might seem a bit pedantic. On the other hand, it is probably a bit more realistic, unless your one bacteria was really meant to represent a large quantity of bacteria.

Anyway, assuming that the number of bacteria is continuous and that you really mean constant relative growth, then your answer seems correct (didn't actually check the numerical at the end, but it's a bit lower than my answer, as it should).

[Lorizael]

Yeah, I did your calculation first (which, of course, doesn't involve any Calculus) but realized that would be too long if there is continual growth.

[Lul Thyme]

Yeah, I did your calculation first (which, of course, doesn't involve any Calculus) but realized that would be too long if there is continual growth.

I'm not sure what you mean.
Which calculation is the right one depends on the problem, it's not a choice.
In the discrete case, if there were many steps, you would use a geometric series.

[Lul Thyme]

The number of units eaten after n hours is :
2^0*60+2^1*60+2^2*60+...+2^(n-1)*60=60(1+2+4+...+2^(n-1))=60(2^n-1).
For example, after 4 hours, 60(2^4-1)=60*15=900 units have been eaten.

[Lorizael]

Yes indeedy. Err, what I'm trying to say is that when this problem popped into my head, my intuitive sense of how to solve it was by doing a geometric series, and that no calculus was actually involved. But I wanted it to be a calculus problem, so I was, in fact, assuming a more steady progression, and not something that happens at the end of every minute and hour. Yes, it would make more sense to have a larger base population if that's what the problem is going to be, but I didn't want to change the problem because I'd already gotten the other answer and wanted to compare to see if the answer I came up with doing it as a calculus problem made sense.

[Tuberski]

Reteaching Calculus?

Why?

ACK!

[Lorizael]

So that I can learn math I was never taught, so that I can learn more science, so that I can conquer the universe.

[SlowwHand]

You can have it.

[Lorizael]

Thanks!

[SlowwHand]

Right. You say that now, but later...

[Lorizael]

Well, thanks for the input, Lul Thyme and Sloww. Now back to learning integration by partial fractions. Because this sonofa***** right here knows how to party on a Saturday night, he does. Especially St. Patrick's Day.

[SlowwHand]

How many nomatical particulates make up the radius of a city block? That should keep you busy.

[Sava]

WHY YO UTRYING TO COCKBLOCK

WAHT DAAAA FCK

i see yo utrying to fkc

you cant urn

[KrazyHorse]

you are too drunk. you are slowing down