Question

A bacterial culture starts with 800 bacteria and grows at a rate proportional to its size. After 6 hours there will be 4800 bacteria.
-Express the population, P(t) after t hours as a function of t

-Using your equation from above, what will the population be after 8 hours?

-Using your equation from above, after how many hours will the population reach
1630 bacteria ?

Answer 1

Answer:

• P(t) = 800·6^(t/6)
• 8722.179
• 2.383 hours

Step-by-step explanation:

Exponential growth can be modeled by ...

  population = (initial population)×(growth factor)^(t/(growth period))

a) Here, our initial population is 800, the growth factor is 4800/800 = 6, and the period for that growth is 6 hours. Then our formula is ...

  P(t) = 800·6^(t/6)

__

It is convenient to graph this to find function values or t values in specific cases. (See below) We can also do that algebraically.

b) P(8) = 800·6^(8/6) ≈ 8722.179

After 8 hours, the population will be 8722.179.

__

c) 1630 = 800·6^(t/6)

  log(1630/800) = (t/6)log(6)

  t = 6·log(1630/800)/log(6) = 2.3833

The population will reach 1630 after about 2.38 hours.

_____

Comment on 3 decimal places

You will notice that our function P(t) uses exact values derived directly from the numbers in the problem statement. You may be expected to put this function in the form ...

  P(t) = p0·e^(kt)

In this case, the value of k is ...

  k = ln(growth factor)/(growth period)

  k = ln(6)/6 ≈ 0.29862658 ≈ 0.299 . . . . .  so P(t) = 800e^(.299t)

If you use e^(0.299t) for computing the other values, you will get different results than shown here. If you maintain the full calculator precision for k, then you should get the same results as shown here.