SOLUTION: The count in a bacterial culture was 400 after 2 hours and 25,600 after 6 hours. We are assuming exponential growth. What is the rate of growth of the population of bact

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Question 135705: The count in a bacterial culture was 400 after 2 hours and 25,600 after 6 hours.
We are assuming exponential growth.

What is the rate of growth of the population of bacteria?
What was the initial population at time t = 0 hours?
Write the function that models the population n(t) after t hours.
When will the number of bacteria exceed 100,000?
Answer by jim_thompson5910(28476) About Me  (Show Source):
You can put this solution on YOUR website!
y=a%2Ab%5Et Start with the general exponential equation


400=a%2Ab%5E2 Plug in t=2 and y=400


400%2Fb%5E2=a Divide both sides by b%5E2 to isolate "a"


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y=a%2Ab%5Et Start with the general exponential equation


25600=a%2Ab%5E6 Plug in t=6 and y=25600


25600=%28400%2Fb%5E2%29%2Ab%5E6 Plug in t=6 and a=400%2Fb%5E2


25600=400b%5E6%2Fb%5E2 Multiply


25600=400b%5E4 Divide by subtracting the exponents


64=b%5E4 Divide both sides by 400



root%284%2C64%29=b Take the fourth root of both sides


b=sqrt%288%29 Simplify






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y=a%2Ab%5Et Go back to the general exponential equation


400=a%2A%28sqrt%288%29%29%5E2 Plug in t=2, b=sqrt%288%29 and y=400


400=a%2A8 Square sqrt%288%29 to get 8


50=a Divide both sides by 8 to isolate "a"





So our equation is y=50%2A%28sqrt%288%29%29%5Et or if you want an approximation of sqrt%288%29 then the equation is y=50%2A%282.82843%29%5Et



a)
"What is the rate of growth of the population of bacteria?"


From the equation, the value of "b" is the rate of growth. So the rate of growth is sqrt%288%29 or 2.82843


b) "What was the initial population at time t = 0 hours?"

y=50%2A%28sqrt%288%29%29%5Et Start with the equation we just found


y=50%2A%28sqrt%288%29%29%5E0 Plug in t=0


y=50%2A%281%29 Raise sqrt%288%29 to the zeroth power to get 1


y=50 Multiply


So the initial population is 50


c) "Write the function that models the population n(t) after t hours."

Earlier we found the equation to be y=50%2A%28sqrt%288%29%29%5Et or y=50%2A%282.82843%29%5Et


d) "When will the number of bacteria exceed 100,000?"


y=50%2A%28sqrt%288%29%29%5Et Start with the equation we just found


100000=50%2A%28sqrt%288%29%29%5Et Plug in y=100000


2000=%28sqrt%288%29%29%5Et Divide both sides by 50


2000=%288%5E%281%2F2%29%29%5Et Rewrite sqrt%288%29 as 8%5E%281%2F2%29


2000=8%5E%28t%2F2%29 Multiply the exponents


log%2810%2C%282000%29%29=log%2810%2C%288%5E%28t%2F2%29%29%29 Take the log of both sides


log%2810%2C%282000%29%29=%28t%2F2%29log%2810%2C%288%29%29 Rewrite the right side


log%2810%2C%282000%29%29%2Flog%2810%2C%288%29%29=%28t%2F2%29 Divide both sides by log%2810%2C%288%29%29


3.65526=t%2F2 Use a calculator to evaluate the left side


7.31052=t Multiply both sides by 2


t=7.31052

So it takes about 7.3 hours for the population to exceed 100,000