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A bacteria culture initially contains 100 cells and grows at a rate proportional to its size. After an hour the population has increased to 570. (a) Find an expression for the number of bacteria after t hours. P(t) = (b) Find the number of bacteria after 4 hours. (Round your answer to the nearest whole number.) P(4) = bacteria (c) Find the rate of growth after 4 hours. (Round your answer to the nearest whole number.) P'(4) = bacteria per hour (d) When will the population reach 10,000? (Round your answer to one decimal place.)

Respuesta :

Answer:

See the explanation.

Step-by-step explanation:

In one hour, the population has increased from 100 to 570.

The population becomes  [tex]\frac{570}{100} \times 100= 570[/tex]% of 100.

Hence, according to our observation, we can tell that the population of the bacteria grows up to 570%  of previous hour.

(a)

The number of bacteria after t hours can be represented as [tex]p(t) = (\frac{570}{100} )^{t} \times 100[/tex].

(b)

After 4 hours, the number of bacteria will be [tex](\frac{570}{100} )^{4} \times 100 = 105560.01[/tex].

(c)

The derivative of p(t) is [tex]100\times ln(5.7) \times (5.7)^t[/tex].

At t = 4, the value will be 183723.6268 ≅ 183724.

(d)

The population will be 10,000 that is p(t) = 10000.

[tex](5.7)^t \times100 = 10000\\(5.7)^t = 100\\t = 2.64594 = 2.6[/tex].