After waiting for hours at the sleazy bar for his superior, Lord Newton Fleming,
ID: 2888283 • Letter: A
Question
After waiting for hours at the sleazy bar for his superior, Lord Newton Fleming, the Spy is about to leave when a courier arrives with shocking news- Lord Fleming has been kidnapped by outside the thug's chateau disguised as an old duck plucker. On the day he arrives, his true identity is known only to the Redselig twins, Hans and Franz. But the next day, Hans' girlfriend, Blabba, finds out. Soon, word begins to circulate among the 60 citizens of the village of the Spy's true identity. He determines that the rate at which the rumor concerning his identity will spread is jointly proportional to the product of the number, N, of those who already know and the number, 60 - N, of those who do Newton Ernst Stavro Blohardt! The Spy hurries to the village dN dt not, so that kn(60-N) for a constant, k. He needs a week to get the information to break into the chateau to free Lord Newton but figures that as soon as 20 or more villagers know his identity, Blohardt is sure to find out too. Does he complete his mission or is he about to be a dead plucker?Explanation / Answer
given dN/dt = kN(60-N)
=>dN/(N(60-N)) = k dt
=>(1/60)((1/N)+(1/60-N))dN = k dt
=>(1/60)((1/N)+(1/60-N))dN = k dt
=>(1/60)(ln(N) -ln(60-N)) = kt +c
=>(ln(N) -ln(60-N)) = 60kt +c
=>ln(N/(60-N))= 60kt +c
=>(N/(60-N))= e60kt +c
=>(N/(60-N))= Ce60kt
=>(60-N)/N= Ce-60kt
=>(60/N)-1= Ce-60kt
=>(60/N)=1+ Ce-60kt
=>(N/60)=1/(1+ Ce-60kt)
=>N=60/(1+ Ce-60kt)
N(0)=2 (2 people hans and franz),N(1) =3 (blabba added)
60/(1+ Ce-60k*0) = 2
=>60/(1+ C) = 2
=>1+C=30
=>C=29
=>N=60/(1+ 29e-60kt)
N(1)=3
=>60/(1+ 29e-60k*1)=3
=>60/(1+ 29e-60k)=3
=>(1+ 29e-60k)=20
=> 29e-60k=19
=> e-60k=19/29
N=60/(1+ 29(19/29)t)
to know time taken for 20 villagers to know the identity
60/(1+ 29(19/29)t)=20
=>(1+ 29(19/29)t)=3
=> 29(19/29)t=2
=>(29/19)t=29/2
=>t =ln(29/2)/ln(29/19)
=>t =6.324
he needs 7 days to get the information
but twenty villagers his identity before that =>blohardt will findout too
so he is about to be a dead plucker
please rate if helpful. please comment if you have any doubt