At t=2 hours, the populations of bacteria colony was 456 and at 4.2 hours, the population was 988. at what time can we expect the population to be 8700?

  10.4 hours

Step-by-step explanation:

It is convenient to write the exponential growth function in these terms:

  population = (initial value)·((value at later time)/(initial value))^exponent

where ...

  exponent = (t - (initial time))/(time difference to later time)

Here, we have ...

So, the exponential function is ...

  p(t) = 456·(988/456)^((t-2)/2.2)

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We can find t for p(t) = 8700 a couple of ways. My favorite is to use a graphing calculator to find t such that ...

  p(t) -8700 = 0

This shows t ≈ 10.4 hours.

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Alternatively, you can solve the equation using algebra.

  8700 = 456·(988/456)^((t-2)/2.2)

  8700/456 = (988/456)^((t-2)/2.2) . . . . . . . . . . .divide by 456

  log(8700/456) = (t -2)/2.2 · log(988/456) . . . . taking logs

  2.2·log(8700/456)/log(988/456) = t -2 . . . . . . multiply by the inverse of the coefficient of t

  t = 2 + 2.2(log(8700/456)/log(988/456)) ≈ 10.389774

We can expect the population to be 8700 at 10.4 hours.

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step-by-step explanation:

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