A scientist has a sample of bacteria that initially contains 10 million microbes. He observes the sample

and finds that the number of bacterial microbes doubles every 20 minutes. Write an exponential equation

that represents M, the total number of bacterial microbes in millions, as a function of t, the number of minutes the sample has been observed. Then, determine how much time, to the nearest minute, will pass

until there are 67 million bacterial microbes.

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M (t) = ?

minutes=?

M (t) = 6.7 * 10⁷ (67 million)

Minutes (t) = 55

Step-by-step explanation:

1. Write an exponential equation that represents M, the total number of bacterial microbes in millions, as a function of t, the number of minutes the sample has been observed.

For answering this question, we will use the following formula:

M (t) = B₀ * g ^ (t/m), where:

M (t) represents the total number of bacterial microbes in millions. B₀ represents the initial population of bacteria in millions. g represents the growth factor. t represents the total number of minutes we will observe the bacteria growing. m represents the time in minutes it takes to the growth factor g to occur.

2. Then, determine how much time, to the nearest minute, will pass until there are 67 million bacterial microbes.

M (t) = B₀ * g ^ (t/m)

Replacing with the values we know:

6.7 * 10⁷ = 10⁷ * 2 ^ (t/20)

6.7 = 2 ^ (t/20) (Dividing by 10⁷ at both sides)

ln 6.7 = ln 2 ^ (t/20)

ln 6.7 = t/20 ln 2

ln 6.7 / ln 2 = t/20

t = ln 6.7/ln 2 * 20

t = 2.74 * 20

t = 54.88

t ≅ 55 (rounding to the nearest minute)