The number of bacteria in a certain culture grows exponentially at a rate of 1% per hour. Assuming that 5,000 bacteria are present initially, find the time required for the bacteria population to reach 45,000. (Round your answer to the nearest hour.)

9514 1404 393

  221 hours

Step-by-step explanation:

The population is given by the exponential equation ...

  population = (initial value) × (1 +growth rate)^t

where the units of t are the same as the units of growth rate.

This lets us write ...

  p(t) = 5000×1.01^t

We want this to be 45000, so ...

  45000 = 5000×1.01^t

  9 = 1.01^t . . . . . . . . . . . . divide by 5000

  log(9) = t×log(1.01) . . . . take logs

  t = log(9)/log(1.01) ≈ 220.8

It will take about 221 hours for the population to reach 45000.

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