Question 237215: This is a story problem I am stuck on could use some help.
A computer is infected with the sasser virus. Assume that it infects 20 other computers in 5 minutes; and that these PCs and servers infect 20 more machines within another five minutes, etc. How long until 100 million computers are infected?
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! 20 computers are infected every 5 minutes.
how long to infect 100 million computers.
this looks like a geometric progression with the common ratio of r = 20
the last term in a geometric progression is given by the formula:
x[n] = x[0] * r^(n-1)
We know x[0] and we know x[n] and we know r.
we need to solve for n.
Our formula becomes:
100 million = 1 * 20^(n-1)
because:
x[0] = 1
r = 20
x[n] = 100 million
this equation becomes:
100 million = 20^(n-1)
we take the log of both sides of this equation to get:
log(100 million) = log(20^(n-1))
by the laws of logarithms, this becomes:
log(100 million) = (n-1)*log(20)
divide both sides of this equation by log(20) to get:
log(100 million) / log(20) = n-1
solve for (n-1) to get:
n-1 = 6.148974295
this means that 100 million computers will be infected in (n-1) * 5 minutes which equals 6.148974295 * 5 = 30.74487147
to see how this works, do the calculations for each n.
0 minutes = 1 * 20^0 = 1
5 minutes = 1 * 20^1 = 20
10 minutes = 1 * 20^2 = 400
15 minutes = 1 * 20^3 = 8000
20 minutes = 1 * 20^4 = 160 thousand
25 minutes = 1 * 20^5 = 3.2 million
30 minutes = 1 * 20^6 = 64 million
30.74487147 minutes = 1 * 20^6.148974295 = 100 million
this can also be solved using the growth formula of:
future = base * e^kt
where e = the scientific constant of 2.718281828....., and k = the constant of growth.
since future = 100 million and base = 1, this formula would become:
100 million = 1 * e^kt
we would need, however, to establish what k is.
since we know that when t = 5, future = 20, our formula would become:
20 = 1*e^5k where:
future = 20
base = 1
k = unknown
t = 5 minutes
we solve this equation for k to get:
20 = e^5k
take natural log of both sides to get:
ln(20) = ln(e^5k)
this becomes:
ln(20) = ln(e) * 5k
this becomes:
ln(20) = 5k
this becomes:
k = ln(20)/5
this becomes:
k = .599146455
our original equation becomes:
100 million = e^(.599146455*t)
take ln of both sides of the equation to get:
ln(100 million) = ln(e^(.599146455*t)
this becomes:
ln(100 million)/.599146455 = t
we solve for t to get:
t = 30.74487147
since this is the same answer we got using the geometric progression, we did good.
the use of the constant "e" simplifies the operation somewhat if you understand that you have to solve for the constant k first, but you can solve it either way and get the same answer.
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