A certain culture initially has 25 bacteria and is observed to double every 5 hours

Question

a) Find an exponential model n(t) = n₀2^t/a for the number of bacteria in the culture after t hours.

b) Estimate the number of bacteria after 18 hours.

c) After how many hours will the bacteria count reach 2 million

Answer 1

a)  n₀ means the initial amount.  In your problem, it tells you that number is 25.  We're told that this culture doubles every 5 hours.  We can use the model given to us in part a to find the variable a.  If it doubles in 5 hours, then the ending amount would be 50 bacteria and the time would be 5 hours.  

50 = 25*2^5/a

Divide both sides by 25 first to get the base with the exponent by itself:  2 = 2^5/a

Because both sides have a base of 2, then the exponents must be the same: 2¹ = 2^5/a

So 1 = 5/a.  Multiply both sides by a and we find that a = 5.

So our exponential model is now n(t) = 25*2^t/5

This will allow you to find the amount of bacteria after any number of hours.

 

b)This is telling us that t is 18.  Just replace t with 18 in the model we found in part a.  Use your calculator to find the final answer.

 

c) This wants us to solve for t if the problem = 2,000,000

So 25 * 2^t/5 = 2,000,000

Divide both sides by 25 first to get the base with the exponent by itself first: 2^t/5 = 80,000

Because we don't have a base of 2 on both sides like we did in Part a, we have to use logs to find the exponent.

Take the log₂ on both sides (need base of 2 since that's the base of the exponent):  log₂ 2^t/5 = log₂ 80,000

On the left, the log "cancels out" the base - on the right, we have to use the "change-of-base formula to put this into a caculator:  t/5 = (log 80,000)/(log 2)

Then multiply both sides by 5 to find t:  t = (log 80,000)/(log 2) * 5

We find that t = 81.44 hours

 

*Note - usually the exponential growth formula is n(t) = n₀2^kt or something similar to that where a rate is multiplied by t rather than divided.  I would suggest that you check that formula first.  The idea and steps listed in this explanation are correct, but the numbers may not be if the formula has been copied incorrectly.