Question

This is a multiple step exponential growth problem, dealing with zombies, help would be amazing!!


Zombies have infected WSU-V after 5 zombies escaped confinement in the science lab at 3:14 AM six hours after their escape there were 72 zombies on campus. Assuming an exponential increase write a function p(h) that represents the population (p) in terms of time in hours (h)

If some one would be willing to walk me through this...
this is just the first part

Answer 1

what method would you like to use? we can use \(e\) or do it an easier way

Answer 2

are you supposed to write something like
\[p(h)=P_0e^{kh}\] ?

Answer 3

it needs to be in terms of an exponential growth function, I'm assuming something along the lines of p(h)=(1+.05)^x? however if we could find an equation that way, it would be just fine

Answer 4

sorry, that would be P=C(1+r)^t I have no idea where i got that other equation from...

Answer 5

ok lets go slow

Answer 6

C being the initial value, r being the rate, and t being time?

Answer 7

so we are not using \(e\) right?

Answer 8

I would presume not, however I am open to ideas seeing as i have attempted this multiple times..

Answer 9

\(C\) is given to you, it is what you start with, namely \(5\)

Answer 10

now \(1+r\) is also easy enough, \(r\) is the increase as a decimal (not a percent)
easiest way to find \(1+r\) is to take \(\frac{72}{5}\) or \(14.4\)

Answer 11

but this takes \(6\) hours, not 1 hour, so this method would give you
\[\large P(h)=5\times (14.4)^{\frac{h}{6}}\]

Answer 12

what is the h/6?

Answer 13

if you want just an \(h\) in the exponent, rather than a \(\frac{h}{6}\) use for your base \((14.4)^{\frac{1}{6}}\) instead of \(14.4\)
you will need a calculator to find it

Answer 14

it is \(\frac{h}{6}\) because it takes 6 hours to increase by a factor of \(14.4\) not one hour

Answer 15

how did you come to h/6 though? what is h? and why does h/6=1?

Answer 16

they asked you to use \(h\) as the variable in the exponent, i.e. the time measured in "hours"
if you want to use \(t \)for time, that is fine too, it is the same thing

Answer 17

oh is it because it took 6 hours for them to reach the current population so that is your reference equation? Then you substitute 6 on top for h or hours?

Answer 18

okay, I think i understand that, so if it were to be 9 hours instead of 6, it would be h/9?

Answer 19

right
i used \(h\) because it said to write \(p(h)\)
yes, exactly!

Answer 20

but if you just want the exponent to be \(h\) rather than \(\frac{h}{6}\) compute
\[14.4^{\frac{1}{6}}\] and use that as your base
then the exponent will just be \(h\)
lets find it

Answer 21

looks like it is about \(1.56\)
http://www.wolframalpha.com/input/?i=%2872%2F5%29^%281%2F6%29

Answer 22

so an alternate answer would be
\[p(h)=5\times (1.56)^h\]

Answer 23

great, thanks! this question also asks me
how many zombies will there be at 4:05 PM?
and
At what time will there be 100,000 zombies? if it is the next day, please indicate it.

what values would you have to substitute in? (that's my own question)

Answer 24

also how did you arrive at 1.56? and why would you use h instead of h/6??

Answer 25

by the law of exponents
\[\large 14.4^{\frac{h}{6}}=(14.4^{\frac{1}{6}})^h\]

Answer 26

so if you want just an \(h\) up in the sky, compute \((14.4)^{\frac{1}{6}}\) and use that as your base

Answer 27

i used a calculator to get it
it really makes no difference, just if for some reason they want the exponent to be \(h\) instead of \(\frac{h}{6}\) that is what you have to do

Answer 28

okay, I think I get that, but how would you find the number of zombies at 4:05 PM? you would have to find the time difference and substitute that for t?

Answer 29

yes, but you would also have to convert the minutes to hours, since your units are hours
kind of a pain, but that is what you need for this

Answer 30

so a 12 hour and 50 minute difference would be a 12 5/6 exponent?

Answer 31

oh right am to pm , didn't catch that
good call

Answer 32

Thanks, almost missed it myself :)
so your equation would be: 5(14.4)^(12 5/6)/6 or just 5(14.4)^12 5/6?

Answer 33

guess you are rounding, that is fine

Answer 34

first one if you are using the base as 14.4

Answer 35

i would convert the mixed number to a fraction first

Answer 36

77/5

Answer 37

actually 77/6

Answer 38

then divide by 6 to get 77/36

Answer 39

right, and i was rounding so that would actually be incorrect, my bad!!

Answer 40

then compute
\[5\times (14.4)^{\frac{77}{36}}\] i don't thing the rounding error is so bad

Answer 41

i get about 1500

Answer 42

why is it 77 over 30?

Answer 43

oh, never mind..

Answer 44

\(36\) not \(30\)
that is because we are using
\[\large p(h)=5\times (14.4)^{\frac{h}{6}}\] with \(h=\frac{77}{6}\)

Answer 45

1513 zombies, if we round up, although i suppose you could have half a zombie...

Answer 46

close enough for government work
not sure how picky your teacher or whoever is

Answer 47

Oh he's very picky, CDO as he likes to say, its OCD with the letters in alphabetical order.

Answer 48

so for the second part (how long until 100,000) you would substitute in values until you get to 100,000?

Answer 49

well then i guess you shouldn't round, even with the minutes
for that you either need to grind it til you find it or use logarithms
depending on if you got to that or not,
i am going to guess not

Answer 50

Well i simply added on a minute so 50 became 51 and that answer should be correct down to the decimal 1512.851644 zombies

Answer 51

k leave it then looks good

Answer 52

although this is actually a test i am re-doing I'll probably end up getting completely different problems tomorrow knowing him.. was i correct in the process for the second part though?

Answer 53

yes, just keep checking
this was on a test?!

Answer 54

Yes, it was, good or bad?

Answer 55

hard i would say, and annoying with hours and minutes etc

Answer 56

It was not fun. At all. and he ruined zombies for me, but thank you so much, I really appreciate your help!

Answer 57

yw