**1) If the graph of a function of the form y = Ce**

^{rt}passes through the points (0,3) and (2,7), determine the constants C and r.

*I started by plugging the value (0,3) into the equation y = Ce*^{rt};

** The input value is 0, which is t, and the output value is 3, which is y. So I just substituted them into the equation**

3 = Ce3 = Ce

^{r0}

*Anything raised to a 0 equals 1 so e*^{r0}= 1.

*3 = C(1)*

*therefore;*

*C=3*

*To solve for r, we start by using C = 3 and the points (2,7) into the equation.*

*7 = 3e*^{r2}

*7/3 = e*^{r2}

*Take the natural log of both sides*

*ln (7/3) = 2r lne*

*We know that ln e = 1. ln e = 1 because ln e=loge e ( I don't know how to make subscripts so bare with me =) ) Base e raised to the power of e is one. This is a rule that applies all the time. I understand what this is about, but I am having difficulty explaining it, so please be free to leave a comment.*

*ex) log5 5=X*

*5*^{x}= 5

*5*^{x}= 5^{1}

*Now back to our question;*

*ln(7/3) = 2r*

*(1/2) ln(7/3) = r*

*so, r = .4236489302 , but we can round it off to four decimal places*

*r = .4236*

**2) Six months after some laboratory rats were delivered to a new research lab, there were 786 rats present. Nine months after the original delivery there were 1720 rats. Assuming exponential growth for the rat population, how many were in the original delivery?**

*The equation we use is P=P*_{o}(model)^{t}**We know,**

**P**

_{6}= 786 t = 6**P**

_{9}= 1720 t = 9

*P*_{0}= ?

*change in t = 9-6*

*= 3*

*We let Po = 786 then t=3. Plug these values into the equation P=P*_{o}(model)^{t}

*1720 = 786 (model)*^{3}

*1720/786 = (model)*^{3}

*Now we use natural log because it deals with population. Populations grow continuously, and that is why we use base e.*

*ln(1720/786) = 3 ln(model)*

*(1/3) ln(1720/786) = ln(model)*

** *remember when using your calculator to find the answer remember to store your answer**

*Sto --> , Aplha A**ln(model) = 0.2610409258 , we can round it to 4 decimal places.*

*model = e ^{.2610}*

*Our equation is P = 786e ^{.2610t}*

*Okay now we found our model now we have to find the original value. There are 2 methods. *

*Method 1:*

*P = 786e ^{.2610t} , but we are looking for six months before so t = -6*

*P _{-6} = 786 e^{.2610(-6)}*

*P _{-6} = 164*

*Method 2:*

*Let P _{6} = 786*

*786 = P _{o}e^{.2610(6)}*

*786/e ^{.2610(6)} = P_{o}*

*164 = P _{o}*

*Homework for tonight is 1.5 in the textbook, odd numbered questions, and to sign up for an account on the websites Mr. K provided in the previous post.*

*This was pretty much what we did today. I hope my work was clear and easy to understand. Please be free to make any corrections if something is unclear. I'll be happy ot fix it. The next person I pick for Scribe is Loso.*

## 9 comments:

HI!

I have to say I think our class has done well in this blogging thing. So far, all entries have been helpful. Use of colours was essential. Just for clarity Ashlynn (and maybe everyone else): for the first question, how did we know where to plug in those points in the equation? Thanks!

The input value is 0, which is t, and the output value is 3, which is y. So I just substituted them into the equation. Does that make sense?

Thanks to Christian for the comment and thanks to Ashlynn for answer the comment. I wasn't so sure about where to plug in the points myself =P This is still a great scribe and I agree with Christian that our blog is going along pretty well and I personally think we have a few HOF worthy posts (one which is already on the HOF, LINGER!) Great job Ashlynn

HI ashlynn,

U pretty much covered everything, but i spotted a little mistake. Isn't it on Method 2 for question 2, the equation is

a.) 786 = Poe.2610(6)

not b.) 786 = Poe.2610(-6)?

its because in a.)if u calculate it, the answer is 164. While in b.) if u calculate it the answer is 3762! Well,sorry to point that out, I just wanna make sure that the blog's published are acurate!

thanks for understanding:

Charlotte;)

thanks to christian and charlotte for pointing those out.

ashlynn, your scribe was simple but easy to understand. everything makes more sense when someone in our class makes a scribe post. =) i feel more relaxed about calculus because of our blog.

good job ashlynn!

be happy.

Pretty colours Ashlynn. I was late in yesterday's class, and what a great example on how the blog helps you stay tuned. Anyway, I just wanted to point out that "base e raised to the power of e is 1," is not correct (i think), it is "base e raised to the exponent of 1, is e".. or I can be completely wrong, or both can be right. But remember, a logarithm is an exponent! =D And also to make subscripts the code is [sub][/sub], pretty ironic hey. Great job!

Hi All,

With all of you working together as you are doing here, how can anyone go wrong! It's wonderful to watch you celebrating, questioning, and assisting each other. A portrait of learning at its best!

Lani

Yes charlotte you were right... I was in a bit of a rush and typed the wrong thing. Manny, I'm not sure about what I wrote, but you may be right. Thats what I heard in class, maybe I heard wrong.

Hi Ashlynn, really nice SP. The colors are nice and easy to see. i would like to point out that for your P6, you should put [sub][/sub] (repla e [] with <>). it makes p knot and the rest that should be subscripted look better =). It's up to you if you want to subscript the letters.

I would like to see ashlynn's sP in the HOF...for use of color, easy to understand, good Sp length

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