Question 1206367


Find the logistic function that satisfies the given conditions.


given:

Initial value = {{{5}}}

limit to growth = {{{20}}}, 

passing through ({{{1}}}, {{{8}}}) 


Use the logistic model:


{{{f(x)=c/(1+a*b^x)}}}

{{{f(0)=5}}}=> at ({{{0}}},{{{5}}})


limit to growth => {{{c=20}}}


{{{5=20/(1+a*b^0)}}}

{{{5=20/(1+a*1)}}}

{{{5=20/(1+a)}}}

{{{5(1+a)=20}}}

{{{5+5a=20}}}

{{{5a=20-5}}}

{{{5a=15}}}

{{{a=3}}}



so far:

{{{f(x)=20/(1+3*b^x)}}}


if passing through ({{{1}}}, {{{8}}}) , we have


{{{8=20/(1+3*b^1)}}}

{{{8=20/(1+3*b)}}}

{{{8(1+3*b)=20}}}

{{{8+24b=20}}}

{{{24b=20-8}}}

{{{24b=12}}}

{{{b=12/24}}}

{{{b=1/2}}}


=> {{{f(x)=20/(1+3*(1/2)^x)}}}


{{{drawing ( 600, 600, -10, 10, -10, 60,
circle(1,8,.13),locate(1,8,p(1,8)),
graph( 600, 600, -10, 10, -10, 60, 20/(1+3(1/2)^x))) }}}