Question

The number of trout t years later is given by P(t) = 700e^ − 0.4t

when t=_______ years there are only 70 trout.

Answer 1

So we have population of trout, `P`, as a function of time, `t`, in years.

Answer 2

They want to know what value of t gives us: \(\Large\bf\sf P(t)=70\)

Answer 3

\[\Large\bf\sf P(t)=700e^{-0.4t}\qquad\to\qquad 70=700e^{-0.4t}\]Understand that step?
We're letting the population equal 70, and we want to figure out which t makes that happen.

Answer 4

Oh and
\(\Large\bf \color{#CC0033 }{\text{Welcome to OpenStudy! :)}}\)

Answer 5

yes I do.. i got to that step

Answer 6

thank you :)

Answer 7

So now we need to solve for t.
It will involve using the natural long.

Answer 8

okay

Answer 9

Let's isolate the exponential first,
we'll start by dividing each side by 700,\[\Large\bf\sf \frac{70}{700}=e^{-0.4t}\qquad\quad\to\qquad\quad \frac{1}{10}=e^{-0.4t}\]

Answer 10

Then we'll take the log of each side,\[\Large\bf\sf \ln \frac{1}{10}=\ln e^{\color{orangered}{-0.4t}}\]Ok our rules of logs let's us do something with \(\Large\bf \color{orangered}{\text{this part}}\).
Do you remember what it is?

Answer 11

i don't remember...maybe it allow us to cancel e?

Answer 12

Mmmm ya that's another way to look at it.
But the rule I was thinking of was this:\[\Large\bf\sf \color{royalblue}{\log(a^b)\quad=\quad b\cdot \log(a)}\]Allows us to bring the exponent out front as a coefficient multiplier.

Answer 13

\[\Large\bf\sf \ln \frac{1}{10}=\color{orangered}{-0.4t}\cdot\ln e\]

Answer 14

And then ln e is simply 1.\[\Large\bf\sf \ln \frac{1}{10}=\color{orangered}{-0.4t}\]

Answer 15

how did you get 1 for e?

Answer 16

Not for e, for ln(e).

Whenever you have a log in which the `base` matches the `contents`, the result is 1.\[\Large\bf\sf \ln (e)\quad=\quad \log_{\color{royalblue}{e}}(\color{royalblue}{e})\quad=\quad 1\]

Answer 17

ohhhhh okay i got it :)

Answer 18

Understand how to solve for t? c:
It'll require some calculator work.\[\Large\bf\sf \ln(0.1)=-0.4t\]

Answer 19

i got -2.305 for In(0.1)

Answer 20

-2.303

Answer 21

then i divide each side by -.04?

Answer 22

yes

Answer 23

5.75

Answer 24

is that correct?

Answer 25

MMmmmmm yah that looks right! :)
So in 5.76 years the population of trout will have fallen to 70.

Answer 26

You can check your work by plugging t=5.76 into your original equation, it should give you a value close to 70

Answer 27

okay! THANKKKKK YOU SO MUCH!! YOUR BETTER THAN MY MATH PROFESSOR!

Answer 28

Haha give him a little credit XD It's hard to give every student personal attention :3

Answer 29

ture! thank you so much!