Question

the data below gives the number of bacteria, in millions, found in a certain culture. CLICKS 2 SEE PLEASE.

Answer 1

Time (hours) -- 0 1 2 3 4

Answer 2

Bacteria ---- 5 8 15 26 48 , consecutively

Answer 3

a. find an exponential function that models the data .
B... write the equation part a in terms of base e.
C. use the model to estimate the doubling time for the culture.

Answer 4

If we have to use an Expoential Model, Think Ratios!

8/5 = 1.6
15/8 = 1.875
26/15 = 1.7333
48/26 = 1.8462

Okay, it's not perfect.

How do you suppose we should build the model? Just guess? Least Squares?

Answer 5

using a formula .

Answer 6

a specific formula ?

Answer 7

Well, yes, we are commanded to use an exponential formula, but how? Which one?

I'm going to guess you're not quite ready for Least Squares.

Exponential formulas can be of all sorts:

\(y=Ae^{bx}\)

\(y=A+Be^{cx}\)

\(y = \dfrac{A}{B + Ce^{dx}}\)

Shall we just pick the simplest?

Answer 8

Sorry, the preview isn't working. It's a little tricky to code the LaTeX to get math expressions.

Answer 9

the simplest, the 1st one ?

Answer 10

Fine. There is still an important choice to make. Since exponentials have CONSTANT ratios, and our data DON'T, what shall we do with that? Any ideas?

Answer 11

make constant ratios out of our data?

Answer 12

Well, we can't just make it be constant when it isn't. The task is to pick a constant ration that approximates the data pretty well. Let's just use the first and last values and see how close we get in the middle.

\(\left(\dfrac{48}{5}\right)^{1/4} = 1.76022\)

See how this values matches NONE of the ratios we first calculated but does sort of hang around them all?

Answer 13

yes .

Answer 14

Well, then let's just go with it.

\(B(t) = 5\cdot 1.76022^{t}\)

Try it out. You shoudl be in the neighborhoos for t = 1, 2, and 3, but right on for t = 0 and 4.

Lastly, let's review to see if we have fulfilled the requirements of the original problem statement.

Answer 15

you want me to give you the values of everything once i place 1,2, and 3, in for t?

Answer 16

You don't have to give them to me. We're trying to convince you that it's a rational model. You need to believe!

Answer 17

for 2 i got 15.491872242

Answer 18

for 3 i got 27.26910335781324

Answer 19

I think you have it. What say you? Are they looking like they are in the neighborhood of the provided values?

Answer 20

yes .

Answer 21

Okay, I think we're done with Part A. How shall we do Part B?

Answer 22

write an equation from what we just got in base of e terms?

Answer 23

That's it. We have to solve this: \(1.76022 = e^{b}\). It is a delightful little logarithm problem. Here's your chance! Use logarithms to sovle for 'b'.

Answer 24

ln (e^ 1.76022) = ln e^b ?

Answer 25

No, there was no 'e' on the left side.