Help! I dont understand how to do this

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 5:00 AM there were 3,000 bacteria present in the culture. At noon, the number of bacteria grew to 3700. How many bacteria will there be at midnight?

Answer 1

This is an example of exponential growth.

 

In this case, P = P₀e^k(t-5), where

P = population of the

P0 = the initial population at 5:00 AM

k = rate of growth

The 5 is here to indicate the time started at 5:00 AM

 

At 5:00 AM, P = 3000 and t = 5

Then 3000 = P₀e^{k(5 -5)} = P₀e⁰ = P₀

P = 3000e^{k(t - 5)}

 

At noon, t = 12 and P = 3700

3700 = 3000e^{k(12 - 5)} = 3000e^7k

e^7k = 37/30

 

Take the log both sides:

7k = ln (37/30)

k = ln(37/30)/7 ≈ 0.03

 

Then P = 3000e^{ln(37/30)(t-5)/7}

 

At midnight, t = 24

Then P = 3000e^{19ln(37/30)/7} ≈ 5300