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

P(t) = 100·2.3^t529 after 2 hours441 per hour, rate of growth at 2 hours5.5 hours to reach 10,000

Step-by-step 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

Lets put the numbers in order...
6,7,7,8,8,9,9,10,11,30

Q1 = 7
Q2 = (8 + 9) / 2 = 17/2 = 8.5
Q3 = 10
IQR = Q3 - Q1 = 10 - 7 = 3 <== so the interquartile range is 3

mean absolute deviation...
the mean of our data is 10.5

now we subtract the mean from every data pointand find its absolute value

6 - 10.5 = -4.5| -4.5| = 4.5
7 - 10.5 = -3.5| -3.5| = 3.5
7 - 10.5 = -3.5|-3.5| = 3.5
8 - 10.5 = - 2.5|-2.5| = 2.5
8 - 10.5 = -2.5|-2.5| = 2.5
9 - 10.5 = -1.5|-1.5| = 1.5
9 - 10.5 = -1.5|-1.5| = 1.5
10 - 10.5 = -0.5..| -0.5| = 0.5
11 - 10.5 = 0.5...|0.5| = 0.5
30 - 10.5 = 19.5..|19.5| = 19.5

now we take the average of these numbers
(4.5+3.5+3.5+2.5+2.5+1.5 + 1.5 + 0.5+0.5+19.5) / 10 =
40/10 = 4 <== ur mean absolute deviation

In summary : 
ur interquartie range (IQR) of ur data set is 3
ur mean absolute deviation (MAD) of ur data set is 4