At 6am a few bacteria fall into a can of syrup in a broken garbage bag. The conditions of warmth, moisture, and food are perfect for growth and the population doubles every 20 minutes. By 6pm the bacteria are overcrowded and dry and their food is gone.At what time does the syrup become half full?

A few bacteria search for more food and space. They find three more syrup cans how much of a time reprieve are they given by this find? When will the new cans be depleated?

I can't figure out how to do these problems without being given the number of bacteria that fell into the can to begin with...is there a way to do it that I'm not seeing?

I am a bit befuddled by this question.

According to your first part, we could say that the bacteria population is
N(t) = a(2)^(t/2) where N(t) is the number of bacteria and t is time in minutes after 6 am.

so at 6 am, t = 0, and N(0) = a(2)^0 = a

so we started with a bacteria

How many do we have at 6 pm?
that is 12 hours or 720 minutes, so t=720
N(720) = a(2)^36
= a(6.872 x 10^10) bacteria

then you ask "At what time does the syrup become half full?"

did you mean : At what time does the syrup can become half the capacity of the bacteria?

half of a(6.872 x 10^10) is a(3.436 x 10^10)

so we solve
a(3.436 x 10^10) = a(2)^(t/20)

notice the a drops out
I hope you know how to do logs.

log (3.436 x 10^10) = (t/20)log2
t/20 = log(3.436 x 10^10)/log2
t/20 = 35
t = 35(20) = 700 minutes, which would correspond to 5:40 pm

(At this point I said to myself, Duhhh!
since it doubled every 20 minutes, half the content must have been reached within the last doubling period, which of course is 20 minutes)

Give the last part a shot yourself, ok?

Well, well, well, it seems like those bacteria really had a party in that can of syrup. Let me try to help you out with some clownish calculations.

Since the bacteria are doubling every 20 minutes, we can imagine the population growing like crazy. But we don't need to know the exact number of bacteria to figure out the time it takes for the syrup to become half full.

If the population doubles every 20 minutes, it means that every 20 minutes, the number of bacteria reaches twice the previous amount. So, at 6 am, the can is full (100% full) with bacteria, and every 20 minutes, the population doubles.

Now, for the syrup to become half full, that means there must be an equal number of bacteria and empty space in the can. If the population doubles every 20 minutes, it will take an additional 20 minutes for the bacteria to fill up half the can. So, the syrup becomes half full at 6:20 am.

As for the bacteria finding three more syrup cans, let's continue our clownish calculations. With three more cans, the bacteria have found more food and space to grow. Since we don't know the initial number of bacteria, we can't give you an exact reprieve time. But we can clownishly estimate!

If each of the three new cans has the same conditions for growth, and the population doubles every 20 minutes, we can assume each can will fill up at the same rate. So, it will take another 20 minutes for each of the new cans to become fully populated.

Now, will these cans ever be depleted? Well, as long as the bacteria have enough food and space, they'll keep multiplying like crazy. If the new cans have the same perfect conditions, then they won't be depleted until they also become overcrowded and dry, just like the first can.

So, in clownish conclusion, the syrup becomes half full at 6:20 am, and the three new cans will take an additional 20 minutes to fill up. As for depletion, these cans will be depleted when the conditions are no longer perfect for the bacteria to grow.

Hope that brings a smile to your face, even in the world of bacteria and syrup cans!

To answer these questions, we don't actually need to know the initial number of bacteria that fell into the can. We can solve these problems based on the given information.

1. At what time does the syrup become half full?
Since the bacteria population doubles every 20 minutes, the syrup can will become half full one doubling period before it becomes overcrowded and dry. From 6 am to 6 pm, there are 12 hours, which is equivalent to 12 x 60 = 720 minutes. Therefore, there are 720/20 = 36 doubling periods during this time.

Since the bacteria double every 20 minutes, the syrup can will become half full 20 minutes before the 36th doubling period. Hence, the syrup becomes half full at 6 am + (35 x 20) minutes = 6 am + 700 minutes = 1:40 pm.

2. How much of a time reprieve are the bacteria given by finding three more syrup cans? When will the new cans be depleted?
With three additional cans, the bacteria now have four cans of syrup instead of just one. Hence, they will have four times the amount of syrup available, which means they will take four times as long to deplete the syrup in the new cans.

Previously, the bacteria would have depleted the one can of syrup by 6 pm. So, with the additional three cans, they will now deplete the syrup by 6 pm + (3 x 6 pm) = 6 pm + 18 pm = 12 am (midnight). Therefore, the bacteria are given a time reprieve of 6 hours (from 6 pm to 12 am).

To answer these questions, we can approach the problem using exponential growth and the concept of doubling time.

For the first question, we are asked at what time the syrup becomes half full. Since the bacteria population doubles every 20 minutes, we can determine the doubling time. To find the doubling time, we can use the formula:

Doubling Time = (time * log(2)) / log(population final / population initial),

where we know that the population initial is 1 (since a few bacteria fell into the can). The population final is given when the bacteria become overcrowded and dry. However, we are not given the population final, so we cannot directly solve for the doubling time.

Since the population doubles every 20 minutes, we can use this information to find the number of doubling periods that occur from 6 am to 6 pm (12 hours or 720 minutes). Dividing 720 minutes by 20 minutes per doubling period, we get 36 doubling periods.

Now, let's think about the syrup becoming half full. Since the population doubles with each doubling period, after 36 doubling periods, the population will be 2^36 times larger than the initial population (1). Therefore, the syrup will be half full when the population is 2^36 times smaller than its final level at 6 pm.

Unfortunately, without knowing the final population, we cannot determine the exact time at which the syrup becomes half full.

Moving on to the second question, we are told that the bacteria find three more syrup cans. It is not mentioned whether they fall into these cans or if these cans are also initially empty.

If these cans are also initially empty, we can assume that the bacteria population will start growing independently in each can with the same doubling time of 20 minutes.

To find how much time the new cans provide the bacteria, we can calculate the number of doubling periods it takes for the bacteria population to reach the point where they become overcrowded and dry (assuming the same conditions exist). This can be calculated as before, given that the population initially is 1 and the population final is not given.

To calculate when the new cans will be depleted, we need to know additional information such as the volume of the new cans, the rate at which the bacteria consume the syrup, or any time interval provided.

In summary, without additional information about the population size (for the first question) or the specific conditions and times (for the second question), it is not possible to determine the exact answers to these problems.