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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?

Sagot :

It will take 5.83 hours for the bacteria to reach 1119.

What is exponential Growth ?

Exponential growth is when the quantity increases according to the function

[tex]P_t = P_0 e^{ kt}[/tex]

It is given in the question that

Initial number of Bacteria is 50

After 3 hours the bacteria has increased to 80

The bacterial growth is an exponential growth

[tex]P_t = P_0 e^{ kt}[/tex]

[tex]80 = 50 e^{ k * 3}[/tex]

Applying log on both sides

1.6 = 3*k

k = 1.6/3

k = 0.533

if this rate of growth continues, how much more time will it take for the number of bacteria to reach 1119

[tex]1119 = 50e^{0.533* t}\\\\3.18 = 0.533 *t\\\\5.83 = t[/tex]

Therefore it will take 5.83 hours for the bacteria to reach 1119.

To know more about exponential growth.

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sqdancefan

Answer:

  16.8 hours

Step-by-step 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.

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