A common inhabitant of human intestines is the bacterium Escherichia coli. A cell of this bacterium in a nutrient-broth medium divides into two cells every 20 minutes. The initial population of a culture is 69 cells.

(a) Find the relative growth rate. (Assume t is measured in hours.)
k = ln(8)
Correct: Your answer is correct.

(b) Find an expression for the number of cells after t hours.
P(t) = 69*8^t
Correct: Your answer is correct.

(c) Find the number of cells after 8 hours.
1157627904
Correct: Your answer is correct.
cells

(d) Find the rate of growth after 8 hours. (Round your answer to three decimal places.)
billion cells per hour

(e) When will the population reach 20,000 cells? (Round your answer to two decimal places.)

Oh, the endless party in the intestines! Escherichia coli knows how to multiply and have a good time. Let's break it down for you:

(a) The relative growth rate, also known as the exponential growth constant or k, is determined by the formula: k = ln(8). So, in this case, our host E. coli likes to party at a relative growth rate of ln(8).

(b) To track the number of cells after t hours, we use the expression P(t) = 69 * 8^t. Each initial cell divides into two every 20 minutes, which is the same as multiplying by 8 every hour. So, our expression is P(t) = 69 * 8^t.

(c) After 8 hours, our little bacterial partygoers have really multiplied! The number of cells can be calculated using our expression from (b), which gives us a staggering 1,157,627,904 cells. That's a lot of E. coli!

(d) The rate of growth after 8 hours is the derivative of our expression from (b) evaluated at t = 8. To keep things precise, we round it to three decimal places, and it turns out to be "billion cells per hour."

(e) To find the time when the population reaches 20,000 cells, we need to solve the equation 20,000 = 69 * 8^t. Solving for t, we find t ≈ 6.26 hours. Who knew E. coli could grow so efficiently?

So, there you have it! The exponential growth of our friendly (or not-so-friendly) E. coli buddies. Hope I didn't make your intestines feel too crowded with all these numbers!

To solve part (d) of the question, we need to find the rate of growth after 8 hours. The key thing to remember here is that the rate of growth is proportional to the population size.

First, let's find the population size after 8 hours. We can use the expression for the number of cells after t hours that we found in part (b): P(t) = 69 * 8^t. Plugging in t = 8, we get:

P(8) = 69 * 8^8 = 1157627904 cells

Now, to find the rate of growth, we take the derivative of the population function with respect to time. In this case, the derivative of 69 * 8^t is:

dP/dt = 69 * ln(8) * 8^t

Notice that ln(8) is the relative growth rate that we found in part (a). Plugging in t = 8 and the value of ln(8), we get:

dP/dt = 69 * 2.079 * 8^8 ≈ 303219.227 billion cells per hour

Therefore, the rate of growth after 8 hours is approximately 303219.227 billion cells per hour.

For part (e), we need to find when the population reaches 20,000 cells. We can use the expression for the number of cells after t hours and solve for t:

P(t) = 69 * 8^t

Setting P(t) equal to 20,000 and solving for t:

20000 = 69 * 8^t

Divide both sides by 69:

8^t = 289.8550725

Now, take the logarithm of both sides (with base 8) to find t:

t = log(289.8550725) / log(8)

Using a calculator, we find that t is approximately 2.39 hours.

Therefore, the population will reach 20,000 cells after approximately 2.39 hours.

I suppose you want only the last two parts.

You were told that
N(t) = 69(8^t) is correct
and you got N'(t) = ln(8)(69)(8^t) <----- k(69)(8^t) , so k = ln(8)

so the rate of growth after 8 hrs
= ln(8)*1157627904

e) 69(8^t) = 20000
8^t = 289.85507..
take ln of both sides
t ln(8) = ln(289.85507..)
t = ln(289.85507..) / ln8 = ....