suppose the population of spiders in a sanctuary grows exponentially. After one day, there were 50 spiders. After three days, there were 200 spiders. How many spiders were there initially?

Question

use x=x₀e^kt equation

Answer 1

50 = xo(e^k))

200 = xo(e^3k)

2 equations 2 unknowns

ln50/xo =k

ln200/xo = 3k

3ln50/xo =3k

3ln(50/xo) = ln(200/xo)

ln((50/xo)^3) = ln(200/xo)

(50/xo)^3 = 200/xo

125000/xo^3 = 200/xo

125,000 = 200xo^2

xo^2 = 125000/200 = 625

xo =sqr625 = 25 spiders on day zero

in one day it doubles to 50

in two days it doubles to 100

in three days it doubles to 200

it's a geometric sequence: 25,50,100,200,400, ...

an =ao(r^(n-1)

a1 =ao(2)^0 = ao = 25

a2 =ao(2)^1 = 25(2) =50

a3 =ao(2)^2 = 25(4) = 100

a4 =ao(2^3) =25(8) = 200