Question

A bacteria culture initially contains 100 cells and grows at a rate proportional to its size. After an hour the population has increased to 230. (a) Find an expression for the number of bacteria after t hours. (b) Find the number of bacteria after 2 hours. (Round your answer to the nearest whole number.) P(2) = ___bacteria (c) Find the rate of growth after 2 hours. (Round your answer to the nearest whole number.) P'(2) = ___bacteria per hour (d) When will the population reach 10,000? (Round your answer to one decimal place.) t = ___hr

Answer 1

• P(t) = 100·2.3^t
• 529 after 2 hours
• 441 per hour, rate of growth at 2 hours
• 5.5 hours to reach 10,000

Explanation:

It often works well to write an exponential expression as ...

value = (initial value)×(growth factor)^(t/(growth period))

(a) Here, the growth factor for the bacteria is given as 230/100 = 2.3 in a period of 1 hour. The initial number is 100, so we can write the pupulation function as ...

P(t) = 100·2.3^t

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(b) P(2) = 100·2.3^2 = 529 . . . number after 2 hours

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(c) P'(t) = ln(2.3)P(t) ≈ 83.2909·2.3^t

P'(2) = 83.2909·2.3^2 ≈ 441 . . . bacteria per hour

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(d) We want to find t such that ...

P(t) = 10000

100·2.3^t = 10000 . . . substitute for P(t)

2.3^t = 100 . . . . . . . . divide by 100

t·log(2.3) = log(100)

t = 2/log(2.3) ≈ 5.5 . . . hours until the population reaches 10,000