Question

a scientist notes that the number of bacteria in a colony is 50.3 hours later, she notes that the number of bacteria has increased to 80. if this rate of growth continues, how much more time will it take for the number of bacteria to reach 1119?

Answer 1

16.8 hours

Explanation:

An exponential population increase can be modeled by the function ...

p(t) = a·b^(t/p)

where 'a' is the initial value (at t=0), b is the multiplier in time period p.

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setup

The colony increased by a factor of b = 80/50 = 1.6 in p = 3 hours. Since we want to find the additional time to reach a population of 1119, the initial population we're working with is 80, not 50.

p(t) = 80·1.6^(t/3)

1119 = 80·1.6^(t/3)

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solution

Solving this for t, we find ...

1119/80 = 1.6^(t/3) . . . . . . . . . . . . divide by 80

log(1119/80) = (t/3)log(1.6) . . . . . take logarithms

t = 3·log(1119/80)/log(1.6) . . . . . divide by the coefficient of t

t ≈ 16.8 . . . . hours

It will take about 16.8 more hours for the population to increase from 80 to 1119.