Question

by natural increase, a city whose population is 40,000 will double in 50 years. there is a net addition of 400 persons per yr because of the people leaving and moving unt the city. estimate the population in 10 yrs. hint: first find the natural growth proportionality factor. Ans: 50, 239

my answer is 49, 965. will post solution below

Answer 1

double time formula:
\[\frac{\ln 2}{k} = t\]
\[\frac{\ln 2}{t} = k\]
\[\frac{\ln 2}{50} = 0.0139\]

\[\ln(\frac{x}{x_o}) = kt\]
\[\frac{x}{x_o} = e^{kt}\]
\[x = x_o e^kt\]
\[x = 40,000 e^{0.0139 \times 10}\]
\[x = 45, 965\]
400 people pr year so in 10 years there are 4,000 people increase
so x = 49, 965

Answer 2

so what did i do wrong

Answer 3

400 * 10 = 4000

this cannot be linear growth

it grows naturally
after 10 years it sums to : 400e^9k + 400e^8k +..... 400e^k + 400

Answer 4

so im going to solve that one by one?

Answer 5

geometric sequence
r = e^-k
n = 10

Answer 6

oh lol of course

Answer 7

remind me what the formula is?

Answer 8

i just googled :
http://regentsprep.org/Regents/math/algtrig/ATP2/GeoSeq.htm

Answer 9

nice thanks

Answer 10

why is r = e^-k?

Answer 11

400e^9k + 400e^8k +..... 400e^k + 400

common ratio : 2nd term/1st term

Answer 12

oh of course...

Answer 13

http://www.wolframalpha.com/input/?i=%28400*e%5E%28.013863%29%28%28e%5E%2810*.013863%29+-+1%29%2F%28e%5E%28.013863%29-1%29%29%29+%2B+40000*e%5E%2810*.013863%29+

Answer 14

im getting that answer... maybe k has to be more significant.. i mean upto 5 or 6 decimal placces

Answer 15

\[s_{10} = \frac{a_1 (1-r^n)}{1-r} \implies \frac{400(1-e^{-0.0139 \times 10})}{1- e^{-0.0139}}\]
is that right?

Answer 16

right

Answer 17

hmmm no wait

Answer 18

no?

Answer 19

a = 400e^9k ...

Answer 20

hmm okay
\[\large S_{10} = \frac{400e^{9 \times 0.0139} (1 - e^{-0.0139 \times 10})}{1 - e^{-0.0139}}\]
correct?

Answer 21

yh looks correct

Answer 22

i got 50, 226

Answer 23

ok.. put the exact value of k in caculator.. maybe we will get the answer given

Answer 24

k = ln2/50

Answer 25

good luck :)

Answer 26

i tried more exact and i got 50, 210

Answer 27

hmmm i think i give up... not getting the book answer :\

Answer 28

lol...i think it's a matter of significant figures...how many do you think i should use

Answer 29

nope.. i have put k = ln2/50

still not getting the book answer

Answer 30

\[40,000\times 2^{\frac{1}{5}}+400\times 10=49,948\] rounded

Answer 31

@satellite73 yes that is what i got too at first but the answer should be 50,239. any idea how that happened?

Answer 32

no

Answer 33

aww okay :(