Question

A population of bacteria is growing according to the function below, where t is in hours. How many hours will it take for the population to grow to 1,000 bacteria?
B(t)=50*e^2.5t

Answer 1

1. 1000= 50 x e^2.5t 
2. 1000/50 = 20 ---> 20 = e^2.5t
3. 20 = e^2.5t ---> ln20 = ln e^2.5 (We take the natural log because it will get rid of e)
4. 2.99 = 2.5t --> 2.99 / 2.5 = 1.198 or 1.20


The answer is 1.198 or approximately 1.20

Answer 2

B(t) = 50 * e^(2.5t)            Grow to 1000

1000 = 50e^(2.5t)
divide both sides by 50
20 = e^(2.5t)
To clear the e, take the LN(natural log) of both sides.
ln 20 = lne^(2.5t)
We know that lne = 1, so we use this rule and the power rule to bring the 2.5t out of the power and in front of the lne.
ln20 = 2.5t * lne
ln20 = 2.5t * 1
ln20 = 2.5t
divide both sides by 2.5
t = 1.2 hours

CHECK
y = 50e^(2.5*1.2)
y = 50 * 20
y = 1000  CHECKS