Question

PLZ HELP!!
Prehistoric cave paintings were discovered in a cave in France. The paint contained 14% of the original Carbon-14. Estimate the age of the paintings.

Answer 1

Hey, Lulu! So this is a half-life problem. Do you konw any equations that can help with this?

Answer 2

\[A=A _{0}e ^{kt}\]

Answer 3

Cool! so you're solving for "t" in this equation. What are the values of the other terms? (hint: you know all of them)

Answer 4

so far i have
\[14=100e ^{-0.00012t}\]

Answer 5

Cool! sorry was just checking...so you have to get t by itself
14 / 100 = e^-0.00012t\

Answer 6

then take ln of both sides to bring that t down. what do you get?

Answer 7

\[\ln 14/100=-0.00012t\]

Answer 8

Yup! then divide by -0.00012 to get t by itself

Answer 9

Nice work! ( http://www.wolframalpha.com/input/?i=solve+ln%2814%2F100%29%3D%E2%88%920.00012t+for+t)

Answer 10

could you help me with another??

Answer 11

Of course!

Answer 12

(This is good practice for me, too...it's been years) :)

Answer 13

ok thank you
An artifact originally had 16 grams of Carbon-14 present. Determine how many grams of Carbon-14 will be present in 11,430 years.

Answer 14

So you'll start with your same formula you had above...but your unknown this time is A

Answer 15

so walk me through how you'd do it! With problems like this I've found it's really good to just list out the variables at the beginning with their values ... or a "?" if you don't know what it is

Answer 16

so .. im pretty sure this is how it would be substituted
\[A=16e ^{-0.00012(11,430)}\]

Answer 17

exactly! Then just plug and chug

Answer 18

i got negative 21.945

Answer 19

Good thing too with problems like these is you can always kind of "Ball park" your answer

Answer 20

i dont think thats right

Answer 21

hahaha, no :) let me check quickly

Answer 22

get rid of that comma? i got the right answer

Answer 23

but BEFORE you fix it - what do you THINK you should get?

Answer 24

a really good way to attack these problems:
t = 11430 years
and the half-life of carbon 14 is something like 5700 years

Answer 25

5730 to be exact

Answer 26

so wht did u get

Answer 27

cool! so before you solve it... if the half life is 5730 years... and it has 11430 years to decay... the carbon's gonna go through almost exactly 2 half-lifes
so you'd expect about 1/4 of the original amount of carbon to be left

Answer 28

4.06g
http://www.wolframalpha.com/input/?i=solve+a%3D16e^%28%E2%88%920.00012%2811430%29%29+for+a

Answer 29

could you help me with a few others??
A bird species in danger of extinction has a population that is decreasing exponentially. Five years ago, the population was at 1400 and today only 1000 of the birds are alive. Once the population drops below 100, the situation will be irreversible. When will this happen?

Answer 30

they are a little different

Answer 31

Hi! I'm back - what's your thought right off the bat on this one?

Answer 32

i know i will use \[A=A _{0}e ^{kt}\]

Answer 33

right, so your issue with this one is figuring out what "k" is

Answer 34

\[100=1000e ^{kt}\]
im not sure what to filll in for t

Answer 35

t is what you're solving for, so that will still be there when you figure out the other values

Answer 36

you're given some informaiton about k though.
you know that in five years... the population fell from 1400 to 1000

Answer 37

sorry i meant k

Answer 38

:) no problem.

Answer 39

yes it fell 400 in 5 years

Answer 40

exactly! so you're basically gonna have to do this twice.

Answer 41

In step 1) you need to figure out k with a KNOWN t (hint hint)
In step 2) you need to plug in that k value to figure out when it gets to 100

Answer 42

is your know t 5

Answer 43

yup!

Answer 44

so you have TWO separate equations:

1) 1000 = 1400e^(k*5)
Figure out k
Then plug into
2) 100 = 1000e^(k*t)

Answer 45

ok

Answer 46

so do you see what we did? we know that it decays a certain amount in a certain time ... so we figure k out from that... to figure out how long it takes to decay some more

Answer 47

i got -16.447
i think i did something wrong

Answer 48

one sec!

Answer 49

from the first eqn.
k = -0.06729
plug it into the second and solve for t
and t= 34.22

Answer 50

ok i understand

Answer 51

it would be 34.22 years?

Answer 52

yup!

Answer 53

ok great
and i was wondering if you could help me with one or two more ( if you dont mind)

Answer 54

The logistic growth function describes the population of an endangered species of birds t years after they are introduced to a non-threatening habitat.
a. How many birds were initially introduced to the habitat?

b. How many birds are expected in the habitat after 10 years?

c. What is the limiting size of the bird population that the habitat will sustain?
\[f(x)=500/1+83.3^{-0.162t}\]

Answer 55

f(x)=500/1+83.3 −0.162t is the equation