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A bacteria culture starts with 2,000 bacteria and doubles in size every 4 hours. Find an exponential model for the size of the culture as a function of time t in hours.

f(t) =

Use the model to predict how many bacteria there will be after 2 days. (Round your answer to the nearest hundred thousand.)
bacteria

User Cheeso
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1 Answer

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Answer:

f(t) = 2000 x 2^(t/4)

After 2 days, there will be approximately 8,200,000 bacteria.

Explanation:

Let's try to build the equation 1 piece at a time. (I also explain how you can use a basic formula to solve this equation in the end, if you want a simple answer first, scroll down to under the 'line' I drew.)

Firstly, the equation needs to double the bacteria culture (starting with 2,000 bacteria). How do we double a number? We multiply it by 2. So, the first part of the equation is:

f(t) = 2000 x 2

f(t) = 2000(2)

Secondly, the question says the bacteria doubles every 4 hours. Let's think about what this means:

Simply put, it means that in four hours, the bacteria will double. So,

  • In 0 hours, the bacteria will stay the same.
  • In 4 hours, the bacteria will double.
  • In 4 x 2 (8) hours, the bacteria will double 2 times.
  • In 4 x 3 (12) hours, the bacteria will double 3 times.

We can use this information to write out what the equation f(t) should be, based on what t is. (t = how many hours have passed)

(It is important that you understand what t means. If t = 0, that means zero hours have passed. If t = 12, that means that 12 hours have passed.)

  • If t = 0, the bacteria will stay the same. So, the equation should give us: 2000
  • If t = 4, the bacteria will double. So, the equation should give us: 2000 x 2
  • If t = (4 x2) = 8 hours, the bacteria will double 2 times. So, the equation should give us 2000 x 2 x 2
  • If t = (4 x 3) 12 hours, the bacteria will double 3 times. So, the equation should give us 2000 x 2 x 2

Okay, we have written out what the equation f(t) should give us based on what t is. Do we notice any patterns? It seems like every time t increases by 4, we multiply 2000 by another 2.

You can go back and look again to see what I mean, notice how:

  • when t=4, we multiply by one 2
  • when t= 4+4 = 8, we multiply by two 2's
  • when t= 4+4+4 = 12, we multiply by three 2's

Now, let's use this pattern to add onto our equation. So far our equation looks like:

f(t) = 2000 x 2

How can we change the amount of 2's in our equation based on what t is (aka, based on how many hours have passed)? We can use exponents:

  • 2²= 2 x 2
  • 2³= 2 x 2 x2
  • Notice how exponents change the amount of 2's being multiplied together

So what exponent can we add to 2 to make the equation f(t) give us the answer we want?

f(t) = 2000 x 2^(t/4)

Why does this work? Well, we can look at what we noticed previously (reread earlier parts if you forgot):

  • when t=4, we want there to be one 2. Our equation does that: 2^(4/4) = 2^(1) = 2
  • when t = 8, we want there to be two 2's. Our equation does that: 2^(8/4) = 2^(2) = 2 x 2
  • when t = 12, we want there to be three 2's. Our equation does that: 2^(8/4) = 2^(2) = 2 x 2 x 2

So, our final equation is: f(t) = 2000 x 2^(t/4)

Now, the question asks how many bacteria there will be after 2 days. In our equation, t represents how many hours have passed, not days. So, we need to figure out how many hours are in 2 days. There are 24 hours in one day, so there are 24x2 = 48 hours in two days.

So, t = 48

(plug 48 into the equation)

f(48) = 2000 x 2^(48/4)

= 8,192,000

= 8,200,000 (rounded to the nearest hundred thousand)

__________________________________________________________

note, this was a very long answer, if you are not interest in understanding how this works, you can use a formula for these types of questions:

f(t) = a x b^(t/c)

a = how many bacteria you start with

b = how much the bacteria increases each time period (so, if the bacteria doubles each time, then b=2. If the bacteria triples, then b=3. If the bacteria halves, then b = 1/2)

c = the time period (how long it takes the bacteria to replicate) (for example, if a bacteria doubles every 10 minutes, then the time period is 10 minutes)

t = the amount of time that has passed (note, c and t must be in the same units. So, if c is in hours, t must be in hours. If c is in minutes, t must be in minutes)

We can easily solve this question by plugging it into this formula given above:

a = 2000

b = 2

c = 4 hours

t = 2 days = 48 hours

f(t) = a x b^(t/c)

f(48) = 2000 x 2^(48/4)

= 8,192,000 = 8,200,000 (rounded)

Let me know if you have any questions!

User Ebrahim Byagowi
by
7.8k points

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