Their population doubled once an hour. At 1:00 a.m. there were 2 bacteria, by 2:00 a.m. there were 4 bacteria, and so forth. At noon, 12 hours later, their food

Question

Imagine a community of bacteria living on a Petri dish. A single bacterium was placed on the dish at midnight. Their population doubled once an hour. At 1:00 a.m. there were 2 bacteria, by 2:00 a.m. there were 4 bacteria, and so forth. At noon, 12 hours later, their food supply ran out. At what time of day was half their food used upImagine a community of bacteria living on a Petri dish. A single bacterium was placed on the dish at midnight. Their population doubled once an hour. At 1:00 a.m. there were 2 bacteria, by 2:00 a.m. there were 4 bacteria, and so forth. At noon, 12 hours later, their food supply ran out. At what time of day was half their food used up

Answer 1

First, find how may bacteria there are after 12 hours and assume that number consumed all the food.  Then assume half of that number consumed half of the food, and calculate at what time you had half of that number.  That time will be your answer.

 

P = population of bacteria, which doubles every hour

 

P(t) = P₀2^t

 

t = hours since midnight

P₀ = initial population = 1 ⇒

 

P(t) = 2^t

 

P(12) = 2¹² = 4096

 

P(12) / 2 = 4096/2 = 2048

 

2048 = 2^t

 

log2048 = t log2

 

t = log2048 / log2 = 11 hours

 

Half the food was assumed 11 hours after midnight.