Question

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.)

Answer 1

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Answer:

221 hours

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.