There will be about __?__ bacteria at midnight.

Question

During a research experiment, it was found that the number of bacteria in a culture grew at a rate proportional to its size. At 10:00 AM there were 2,000 bacteria present in the culture. At noon, the number of bacteria grew to 2,600. How many bacteria will there be at midnight?

Answer 1

GREETINGS FELLOW FLORIDIAN!!!!!

 

The model is exponential f(x) = k* E^(cX) = k*exp(cX) where x is the hour.

 

 So beginning at 10 am, X=0

                        11 am, X=1

                        12 noon,X=2  <-- 2 hours have elapsed

                        1 pm , X=3

                        2 pm, X=4

                         ...etc etc...                      

 

                         so midnight is X=14 hours later

 

The following two facts are given in the problem.

f(0) = 2000 and f(2) = 2600

'

Plugging the first fact into the equation gives:  2000 = f(0) = k*exp(C*0) = k

 so the model is now f(x) = 2000*exp(CX)

 

 Plugging the second point into the function model gives:

   2600 = f(2) = 2000*exp(2C)

 

  Dividing both sides by 2000 and cancelling a factor of 100, the equation becomes:

 

  13/10 = exp(2C)

 

 Taking the log of both sides:

 

  ln(13/10) = 2C

 

 so C = ln(13/10)/2

 

The final , complete function model is f(x) = 2000*EXP((LN(13/10)/2*X))

 

The formula was copied and pasted from MS Excel and it works.

 

At midnight, or f(14) = 12549.7

You can say 12550

 

 

Here's the spreadsheet:

hour  vs population

0 2000
1 2280.35085
2 2600
3 2964.456105
4 3380
5 3853.792937
6 4394
7 5009.930818
8 5712.2
9 6512.910063
10 7425.86
11 8466.783082
12 9653.618
13 11006.81801
14 12549.7034
15 14308.86341
16 16314.61442
17 18601.52243
18 21208.99875
19 24181.97916
20 27571.69837