The populations of termites and spiders in a certain house a

The populations of termites and spiders in a certain house are growing exponentially. The house contains 110 termites the day you move in. After four days, the house contains 205 termites. Three days after moving in, there are two times as many termites as spiders. Eight days after moving in, there were four times as many termites as spiders. How long (in days) does it take the population of spiders to triple? () _____________ days

Solution

let amount of termites after time t(in days) is y and amount of spiders after time t(in days) is x

for termites :general exponential equation is y=a*b^t

a=110

y=110*b^t

at t=4, y=205

205=110*b⁴

=>b=1.1684

y=110*1.1684^t

for spiders :general exponential equation is x=c*d^t

at t=3, y=2x

110*1.1684³=2c*d³

at t=8, y=4x

110*1.1684⁸=4c*d⁸

(4c*d⁸)_/(2c*d³)=(110*1.1684⁸)_{/(110*1.1684³)}

2d⁵=1.1684⁵

d=1.0887332

110*1.1684³=2c*d³

110*1.1684³=2c*1.0887332³

=>c=67.978

so,x=67.978*1.0887332^t

population of spider to triple

3*67.978=67.978*1.0887332^t

1.0887332^t=3

t=12.9

it takes 12.9 days